# Affine Deformations of Divisible Convex Cones and Affine Spacetimes

Antoine Ablondi

<figure><img src="image_1.png" /></figure>

## To cite this version:

Antoine Ablondi. Affine Deformations of Divisible Convex Cones and Affine Spacetimes. 2025. hal-05153553

HAL Id: hal-05153553

https://hal.science/hal-05153553v1

Preprint submitted on 9 Jul 2025

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution 4.0 International License
# AFFINE DEFORMATIONS OF DIVISIBLE CONVEX CONES AND AFFINE SPACETIMES

ANTOINE ABLONDI

ABSTRACT. Given a subgroup of SL($\mathbb{R}^{d+1}$) $\times \mathbb{R}^{d+1}$ obtained by adding a translation part to a torsion-free discrete subgroup of SL($\mathbb{R}^{d+1}$) dividing a convex domain in the sense of Benoist, we study two particular convex domains on which the affine action of that subgroup is free and properly discontinuous and whose quotients are naturally endowed with an "affine spacetime" structure that we introduce. In that setting, we show that those quotients are Maximal Globally Hyperbolic affine spacetimes admitting a locally uniformly Convex and Compact Cauchy surface (denoted as a MGHCC affine spacetimes), and that they come with a cosmological time function with Cauchy hypersurfaces foliating the quotient affine spacetime as level sets. Finally, we show that those are the only examples of such MGHCC affine spacetimes. All these results generalise the work of Mess, Barbot and Bonsante on affine deformations of uniform lattices of $SO_0(d, 1)$.

## CONTENTS

<table><tr><td>1. Introduction</td><td>1</td></tr><tr><td>2. Affine spacetimes</td><td>5</td></tr><tr><td>3. Backgrounds</td><td>9</td></tr><tr><td>4. C-convex domains in the affine spacetime (A<sup>d+1</sup>, C)</td><td>13</td></tr><tr><td>5. Affine deformations of divisible convex cones</td><td>19</td></tr><tr><td>6. Cosmological time on C-convex domains</td><td>26</td></tr><tr><td>7. Quotient of the maximal domain</td><td>35</td></tr><tr><td>8. Correspondence between affine deformations of dividing groups and affine spacetimes</td><td>37</td></tr><tr><td>Appendix A. Examples of dividing groups admitting affine deformations</td><td>42</td></tr><tr><td>Appendix B. The projective approach to affine deformations</td><td>44</td></tr><tr><td>References</td><td>46</td></tr></table>

## 1. INTRODUCTION

In this article we study some particular discrete subgroups of $SA(\mathbb{R}^{d+1}) = SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$, the group of special affine transformations of $\mathbb{R}^{d+1}$. Our interest for such groups comes from previous works in two different settings generalising uniform lattices of $SO_0(d, 1)$.

On one side, affine deformations of uniform lattices of $SO_0(d, 1)$, that is subgroups of $SO_0(d, 1) \times \mathbb{R}^{d+1}$ obtained by adding translation parts to uniform lattices, were the focus of the works of Mess [Mes07, ABB+07] for $d = 2$, as well as the ones of Barbot [Bar05] and Bonsante [Bon05] in higher dimension. They proved a correspondence between those subgroups and some flat Lorentzian spacetimes satisfying nice properties. More recent works [FV16, BF17, FS19, BF20] have shown that tools from convex geometry inside the Minkowski space $\mathbb{R}^{d,1}$ (on which flat Lorentzian spacetimes are locally modelled) were very useful in the study of those objects, and that their Lorentzian metric was not as important as their causal and affine structures.

On the other side, the work of Benoist on the rich topic of divisible convex sets [Ben03, Ben04, Ben05, Ben06] offers an affine generalisation of uniform lattices of $SO_0(d, 1)$: by considering discrete torsion free subgroups of $SL(\mathbb{R}^{d+1})$ preserving an open proper convex cone $C \subset \mathbb{R}^{d+1}$ and with properly discontinuous and cocompact projective action on $\mathbb{P}(C)$, the projectivisation of the cone.
**Main results.** In this article we introduce and study affine spacetimes, which are a natural generalisation of time oriented flat Lorentzian spacetimes. Indeed, we shall define a $(d+1)$-dimensional affine spacetime $(M, C)$ as a manifold $M$ endowed with both a $(G, X)$-structure where $G = \text{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ is the group of orientation and volume preserving affine transformations acting on the affine space $X = \mathbb{R}^{d+1}$, and a parallel distribution of open proper convex cones $C$ in its tangent bundle. Thus, time oriented flat Lorentzian spacetimes are naturally affine spacetimes: they are locally modelled on the Minkowski space $\mathbb{R}^{d,1}$, so they admit a $(\text{SO}_0(d, 1) \times \mathbb{R}^{d,1}, \mathbb{R}^{d,1})$-structure and a natural cone distribution given by the interiors of the future light cones in its tangent bundle. All of the definitions and notions arising purely from the causal structure of Lorentzian spacetimes, for example *Cauchy surfaces* and *global hyperbolicity*, can naturally extend to this new setting (see subsection 2.3). Hence, affine spacetimes are locally modelled on the *model affine spacetime* $(\mathbb{A}^{d+1}, C)$: the real affine space $\mathbb{A}^{d+1}$ endowed with a parallel cone structure given by an open proper convex cone $C$. In that setting, we introduce $C$-convex domains of $(\mathbb{A}^{d+1}, C)$ as open convex domains which are their own $C$-future and with Gauss map satisfying some surjectivity condition (see Definition 4.2). We then prove the following theorem generalising the work of Mess [Mes07, ABB$^{+}$07], Barbot [Bar05], and Bonsante [Bon05].

**Theorem 1.1.** Let $\Gamma_{\tau} < \text{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ be a discrete torsion-free group obtained by adding translations part to a subgroup $\Gamma < \text{SL}(\mathbb{R}^{d+1})$ dividing an open proper convex cone $C \subset \mathbb{R}^{d+1}$, i.e. such that the projective action of $\Gamma$ on $\mathbb{P}(C)$ is properly discontinuous and cocompact. There exists a unique maximal $\Gamma_{\tau}$-invariant $C$-convex domain $D_{\tau} \subset \mathbb{R}^{d+1}$. It satisfies the following properties:

(1) The action of $\Gamma_{\tau}$ on $D_{\tau}$ is free and properly discontinuous, and there is a homeomorphism

$$D_{\tau}/\Gamma_{\tau} \simeq \mathbb{P}(C)/\Gamma \times \mathbb{R}.$$

(2) The affine spacetime $(D_{\tau}/\Gamma_{\tau}, C)$ is a maximal globally hyperbolic spacetime admitting a locally uniformly convex and compact Cauchy surface (a MGHCC affine spacetime).

*Remark.* The existence and uniqueness of the maximal $\Gamma_{\tau}$-invariant $C$-convex domain $D_{\tau}$, on which $\Gamma_{\tau}$ acts properly discontinuously was already proved in the work of Choi [Cho25, Theorems 4.3.1 and 4.3.7] in a more general setting and with different language and approach. In dimension $2+1$, that, as well as the topology of the quotient $D_{\tau}/\Gamma_{\tau}$, was also independently shown by Nie and Seppi [NS23, Theorem A]. The topology of the quotient in higher dimension, the notion of affine spacetime, the whole point (2), as well as Theorems 1.2 and 1.3 below are completely new to the best of our knowledge.

Notice that Theorem 1.1 can also be applied using the cone $-C$, which is also divisible by $\Gamma$, in order to produce another couple of domain and affine spacetime.

Applying Theorem 1.1 in the particular case where $\Gamma < \text{SO}_0(d, 1)$ is the holonomy of a closed hyperbolic manifold and $C$ is the future light cone in the Minkowski space $\mathbb{R}^{d,1}$, we recover some of the works of Mess [Mes07, ABB$^{+}$07], Barbot [Bar05], and Bonsante [Bon05].

In order to complete our generalisation of the correspondence of Mess, Barbot, and Bonsante, we also show the following theorem, stating that these examples of MGHCC affine spacetimes given by Theorem 1.1 happen to be the only ones.

**Theorem 1.2.** Let $(M, C^M)$ be MGHCC affine spacetime. Any pair $(\text{dev}, \rho)$ of a developing map

$$\text{dev} : \tilde{M} \to \mathbb{R}^{d+1},$$

and a holonomy map

$$\rho : \pi_1 M \to \text{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1},$$

satisfies the following:
• $\rho$ is a group isomorphism onto its image $\rho(\pi_1 M)$,

• $\rho(\pi_1 M)$ is an affine deformation $\Gamma_\tau$ of its linear part $\Gamma < \text{SL}(\mathbb{R}^{d+1})$ which divides the cone $C := \text{d} \text{dev}(C^M)$,

• $\text{dev}$ is an isomorphism of affine spacetimes onto its image which is $(D_\tau, C)$, where $D_\tau$, the domain associated with $\Gamma_\tau := \rho(\pi_1 M)$ and $C$, is given by Theorem 1.1.

**Affine cosmological time.** Affine spacetimes significantly differ from flat Lorentzian spaces as their cone distribution does not come as the distribution of isotropic cones for a Lorentzian metric. Nevertheless, similarly to recent works of Kunziger and Clemens [KS18], Papadopoulos and Yamada [PY19], or Buro [Bur23], based on the axiomatisation of timelike spaces by Busemann [Bus67], one can introduce a way to define length on the family of causal curves, satisfying, among other good properties, a time inequality. Then, a cosmological time function (see Section 6) can be defined similarly to the usual Lorentzian one [AGH98]. We then prove the following fact concerning $C$-convex domain of the model affine spacetime $(\mathbb{R}^{d+1}, C)$.

**Theorem 1.3.** Let $C$ be an open proper convex cone in $\mathbb{R}^{d+1}$ with vertex at the origin. Every $C$-convex domain $K$ in $(\mathbb{R}^{d+1}, C)$ admits a concave $C^1$ cosmological time function $T_K : K \to \mathbb{R}_+^*$ whose level sets are $C^1$ convex Cauchy surfaces of $(K, C)$ foliating $K$. Moreover, $T_K$ is invariant by any transformation of $\text{SA}(\mathbb{R}^{d+1})$ preserving $K$.

The cosmological time and its foliation will be a central tool in order to study topological aspects of the action of $\Gamma_\tau$ on $D_\tau$ and prove Theorem 1.1.

The cosmological time on a domain $D_\tau$ from Theorem 1.1 is thus $\Gamma_\tau$-invariant and descends to the quotient, so that the affine spacetime $(D_\tau/\Gamma_\tau, C)$ is foliated by the level hypersurfaces of its cosmological time, which are $C^1$ embedded convex Cauchy surfaces. Using Theorems 1.1, 1.2, and 1.3, that gives the following.

**Corollary 1.4.** Any MGHCC affine spacetime $(M, C)$ is naturally endowed with a $C^1$ locally concave cosmological time $T$. The level hypersurfaces of this cosmological time are $C^1$ $C$-spacelike convex Cauchy surfaces foliating $M$.

In this article we choose to use Cheng-Yau affine spheres [CY86] in order to normalise timelike vectors, measure lengths of timelike curves, and produce the cosmological time function from Theorem 1.3. That choice is partially arbitrary: one could use any other known locally convex hypersurfaces invariant by automorphisms of the cone and with similar properties (strict convexity, smoothness, asymptotic behaviour), for instance hypersurfaces given by the level sets of the Vinberg functional [Vin63, Gol22]. Still, one can notice that in the case of a homogeneous convex cone (with automorphism group acting transitively), which happens to be the setting of the Koszul-Vinberg theory [Kos62, Vin63], up to homothety there is a unique hypersurface satisfying those properties (hence the affine sphere and the Vinberg hypersurface coincide). One of the aims of this article is thus to show how useful and natural is the use of affine spheres and the theory of affine differential geometry in that setting.

On the one side, that choice to normalise vectors through affine spheres is an actual generalisation of the Lorentzian case: in any tangent space of a Lorentzian spacetime, timelike vectors are normalised by the hyperboloid which is a Cheng-Yau affine sphere associated with the isotropic cone of the Lorentzian metric. The work of Minguzzi [Min17] actually gives a physical interpretation to that choice. Moreover, affine spheres have the advantage to behave well with the duality and convexity tools that were used in the Lorentzian case. Indeed, fundamental tools from convex analysis, such as duality, subdifferentials or the Legendre-Fenchel transform, were already used by Cheng and Yau [CY77, CY86] in order to prove their fundamental theorem about existence of complete hyperbolic affine sphere asymptotic to any given proper convex cone in the affine space.
On the other side, tools from convex analysis and convex bodies theory [Sch93, Fil13] have been successfully used in the study of flat Lorentzian spacetimes [FV16, BF17], through the description and use of co-Minkowski geometry [FS19, BF20], also known as “half-pipe” geometry [Dan13]. Those tools rely mainly on the study of variations of the volume of convex bodies and properties of their Gauss map. The affine space, which is devoid of metrics but happens to be endowed with a volume form, is a perfect setting to extend those works, and affine spheres are natural candidates to be images of Gauss maps. Substituting those hyperbolic affine spheres for the sphere or the hyperboloid in, respectively, Euclidean or Minkowski convex geometry [Sch93, Fil13], one can also define and study an affine version of the Minkowski problem, generalising results in the Minkowski space [BF17], but also results about surfaces with constant affine Gaussian curvature [Lab07, NS22]. We leave this problem for future investigation [Abl].

**Links with higher Teichmüller theory.** Let us first focus on the particularly interesting case where $d=2$. Let $S_g$ be a closed oriented and connected surface of genus $g \ge 2$. We consider $\mathcal{P}_g$ the moduli space of convex projective structures on $S_g$. By the work of Choi and Goldman [CG93], $\mathcal{P}_g$ is identified with the Hitchin component [Hit92] of the representation space of $\pi_1(S_g)$ into $SL(\mathbb{R}^3)$ and it can be described as

$$ \mathcal{P}_g = \mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3)) / \mathrm{SL}(\mathbb{R}^3), $$

where the quotient is by conjugation and $\mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3))$ is the set of faithful group representations $\rho \in \mathrm{Hom}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3))$ such that there is an open proper convex cone $C_\rho \subset \mathbb{R}^3$ divisible by $\rho(\pi_1(S_g))$. Then, similarly, $\widehat{\mathcal{P}}_g$, the moduli space of affine deformations of convex projective structures on $S_g$, can be described as

$$ \widehat{\mathcal{P}}_g = \mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3) \times \mathbb{R}^3) / (\mathrm{SL}(\mathbb{R}^3) \times \mathbb{R}^3), $$

where the quotient is by conjugation and $\mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3) \times \mathbb{R}^3)$ is the set of group representations $\rho_\tau \in \mathrm{Hom}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3) \times \mathbb{R}^3)$ with linear part $\rho \in \mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3))$.

Labourie [Lab07] and Loftin [Lof01] both independently used Choi-Goldman theorem to get a holomorphic parametrisation of those Hitchin components. More recently, Nie and Seppi [NS22] have highlighted how $\widehat{\mathcal{P}}_g$ is naturally a topological vector bundle of rank $6g-6$ on the $(16g-16)$-dimensional Hitchin component given by the natural projection

$$ \left\{ \begin{array}{ccc} \mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3) \times \mathbb{R}^3) & \to & \mathrm{Hom}_{\mathrm{div}}(\pi_1(S_g), \mathrm{SL}(\mathbb{R}^3)) \\ \rho_\tau & \mapsto & \rho \end{array} \right. $$

Through the special identification between the Lie algebra $\mathfrak{so}(2,1)$ and the Minkowski space $\mathbb{R}^{2,1}$, affine deformations of Fuchsian subgroups of $\mathrm{SO}_0(2,1)$ can be seen as first order derivatives of variation of hyperbolic structures on $S_g$. Then the part of $\widehat{\mathcal{P}}_g$ over the Teichmüller space $\mathcal{T}_g \subset \mathcal{P}_g$, studied by Mess [Mes07], can be identified with the tangent bundle of $\mathcal{T}_g$. Theorem 1.1 and Theorem 1.2 thus give a correspondence between each element of $\widehat{\mathcal{P}}_g$ and two (maybe equal) isomorphisms classes of $(2+1)$-dimensional MGHCC affine spacetimes with Cauchy surfaces homeomorphic to $S_g$. In the Minkowski case [Mes07, Bon05], one can make this correspondence one to one by arbitrarily choosing a time orientation on the Minkowski space. In the general case, as the divisible cones vary, one cannot make such a simple choice.

In [Lab07, Section 7], Labourie explicitly gives two different parametrisations of the moduli space $\widehat{\mathcal{P}}_g$, as well as some symmetries on it. Using the correspondence presented in this article, and a geometric study of the $(2+1)$-dimensional MGHCC affine spacetimes associated to elements of $\widehat{\mathcal{P}}_g$, we can hope for new geometric counterparts and explanations to Labourie's work (similarly to previous works on the Teichmüller case [BS16, BF20]).
The work of Benoist [Ben05] also has highlighted that divisibility gives a geometric interpretation to some connected components of higher moduli spaces of representations (higher in the sense that they have images in higher rank Lie groups, such as $SL(\mathbb{R}^{d+1})$, but also as they concern representations of fundamental groups of manifolds of higher dimension $d \ge 3$). Letting $\Gamma_0$ be a finitely generated group, he considered $\text{Hom}_{\text{div}}(\Gamma_0, \text{SL}(\mathbb{R}^{d+1}))$ the set of faithful group representation $\rho \in \text{Hom}(\Gamma_0, \text{SL}(\mathbb{R}^{d+1}))$ such that there is an open proper convex cone $C_\rho \subset \mathbb{R}^{d+1}$ divisible by $\rho(\Gamma_0)$. He proved that if $\text{Hom}_{\text{div}}(\Gamma_0, \text{SL}(\mathbb{R}^{d+1}))$ admits a strongly irreducible representation then it consists of a union of connected components of $\text{Hom}(\Gamma_0, \text{SL}(\mathbb{R}^{d+1}))$.

Adopting a projective approach, we shall also highlight that the discrete subgroups $\Gamma_\tau$ considered in Theorem 1.1 have a *convex cocompact action* on an open subset of $\mathbb{P}(\mathbb{R}^{d+2})$ (see Appendix B). When $\Gamma$ is *word hyperbolic*, those are moreover *strongly convex cocompact* in the sense of Danciger, Guéritaud, and Kassel [DGK24], and thus, following from their work, are also *projective Anosov*.

It is compelling to try to link our approach to the recent work of Bobb and Farre [BF24] concerning the generalisation of the part of the Mess correspondence concerning measured geodesic laminations. It should be possible to make those works meet using convex geometry and area measures associated with *C*-convex domain, similarly to what was done in the Minkowski case [FV16, BF17, BF20].

**Organisation of the article.** We first introduce the definition of an affine spacetime in Section 2. We then give a few useful results and definitions from affine differential geometry and convex analysis in Section 3. Section 4 is dedicated to the study of the special case of the model affine spacetime $(\mathbb{A}^{d+1}, \mathcal{C})$: the real affine space $\mathbb{A}^{d+1}$ endowed with a parallel cone structure given by an open proper convex cone $\mathcal{C}$. In Section 5, we focus on the case where $\mathcal{C}$ is divisible by a subgroup $\Gamma$ of $\text{SL}(\mathbb{V}^{d+1})$ the group of special linear transformations of the vector space $\mathbb{V}^{d+1}$. In that setting, we build for any affine deformation of $\Gamma$ the maximal invariant domain $D_\tau$ from Theorem 1.1. Within Section 6, we introduce and study a cosmological time function on $\mathcal{C}$-convex domains, as well as giving it a geometric interpretation in the context of pseudo-Finsler geometry. That allows us, in Section 7, to study the spacetime structure of the quotients $D_\tau/\Gamma_\tau$ and prove point (1) and the GHCC part of point (2) of Theorem 1.1, as well as Theorem 1.3. Finally, Section 8 consists of the proofs of the maximality part of point (2) in Theorem 1.1 and of Theorem 1.2, thus extending the Mess correspondence. In addition, Appendix A gives examples in any dimension of groups dividing an open proper convex domains not projectively equivalent to an ellipse (giving the Klein ball model of the hyperbolic space), and admitting non-trivial affine deformations, while Appendix B links the work of this article to the notion of convex cocompact representation through a more projective approach to affine deformations.

**Acknowledgements.** I am deeply grateful to my two doctoral advisors François Fillastre and Andrea Seppi for introducing me to this subject, helping me in my mathematical work and guiding me through writing this article. I would like to thank Thierry Barbot for interesting discussions about affine spacetimes, Martin Bobb and James Farre for introducing me to the projective approach described in Appendix B, and Colin Davalo for his helpful indications concerning Lemma 5.2. I also thank Suhyoung Choi for noticing and explaining me how some of the results presented in this article are linked to his work.

## 2. AFFINE SPACETIMES

Let us first introduce and study one of the central notions of this work: affine spacetime.

2.1. **Equiaffine manifolds.** Let $d \ge 2$ be an integer. We shall mostly work with $\mathbb{R}^{d+1}$ and use different notations to denote that space, depending on the underlying structures under consideration:

* let $\mathbb{V}^{d+1}$ denote the $(d+1)$-dimensional real vector space endowed with the volume form given by the determinant map $\det : \Lambda^{d+1} \mathbb{V}^{d+1} \to \mathbb{R}$,
• let $\mathbb{A}^{d+1}$ denote the $(d+1)$-dimensional real affine space with associated vector space $\mathbb{V}^{d+1}$, it is endowed with the volume form from $\mathbb{V}^{d+1}$, the flat affine connection $D$ and an orientation.

Throughout the whole article, as long as we have not made a choice of parametrisation we shall use the notations $\mathbb{A}^{d+1}$ and $\mathbb{V}^{d+1}$ rather than $\mathbb{R}^{d+1}$.

A real equiaffine structure, or simply equiaffine structure, on a smooth manifold $M$ of dimension $d+1$, consists of an atlas with values in the affine space $\mathbb{A}^{d+1}$, such that the transition maps are restrictions of orientation and volume preserving affine transformations of $\mathbb{A}^{d+1}$. In the language of $(G, X)$-structures, an equiaffine structure on $M$ is a $(\mathrm{SA}(\mathbb{A}^{d+1}), \mathbb{A}^{d+1})$-structure, where $\mathrm{SA}(\mathbb{A}^{d+1})$ is the group of special affine transformations of $\mathbb{A}^{d+1}$, namely:

$$ \mathrm{SA}(\mathbb{A}^{d+1}) = \mathrm{SL}(\mathbb{V}^{d+1}) \ltimes \mathbb{V}^{d+1}. $$

A manifold endowed with an equiaffine structure is called an *equiaffine manifold*. It is naturally endowed with an orientation, a flat torsion-free affine connection (coming from the flat connection on $\mathbb{A}^{d+1}$) and a volume form parallel under that connection.

If $M$ is an equiaffine manifold with universal cover $\widetilde{M}$, the theory of $(G, X)$-structures tells us that one can fix a *developing map*

$$ \mathrm{dev} : \widetilde{M} \to \mathbb{A}^{d+1} $$

and a *holonomy representation*

$$ \rho : \pi_1 M \to \mathrm{SA}(\mathbb{A}^{d+1}), $$

satisfying the following:

• the developing map $\mathrm{dev}$ is $\rho$-equivariant, in the sense that for all $p \in \widetilde{M}$ and $\gamma \in \pi_1 M$,

$$ \mathrm{dev}(\gamma \cdot p) = \rho(\gamma) \mathrm{dev}(p), $$

• the developing map $\mathrm{dev}$ is a local diffeomorphism preserving the equiaffine structure, i.e. on each chart of the equiaffine structure of $\widetilde{M}$, $\mathrm{dev}$ is an element of $\mathrm{SA}(\mathbb{A}^{d+1})$.

Moreover, given two such pairs $(\mathrm{dev}, \rho)$ and $(\mathrm{dev}', \rho')$ there exists a unique $g \in \mathrm{SA}(\mathbb{A}^{d+1})$ such that

$$ \mathrm{dev}' = g \mathrm{dev} \quad \text{and} \quad \rho' = g \rho g^{-1}. $$

Equivalently, without the language of $(G, X)$-structures, an equiaffine manifold is just a triplet $(M, D, \omega)$ where $M$ is an orientable manifold endowed with a flat torsion-free affine connection $D$ and a volume form $\omega$ such that $D\omega = 0$.

## 2.2. Cone structure and causality.

**Definition 2.1** (Convex cone structure). A convex cone structure $\mathcal{C}$ on a smooth connected manifold $M$ is a field of cones $\{C_p | p \in M\}$, where for every point $p \in M$, $C_p$ is an open proper (i.e. non-empty and having closure $\overline{\mathcal{C}}$ containing no entire line) convex cone in $T_p M$ with vertex at the origin.

The following notions are classical in Lorentzian geometry. Noticing that they rely only on the existence of a light cone distribution in the tangent bundle and not on the metric, we can naturally extend those to our setting.

**Definition 2.2** ($\mathcal{C}$-spacelike, $\mathcal{C}$-null and $\mathcal{C}$-timelike vector). Let $M$ be a smooth connected manifold endowed with a convex cone structure $\mathcal{C}$. A non-zero vector $v \in T_p M$ is said to be:

• $\mathcal{C}$-spacelike if $v \notin \overline{C_p} \cup (-C_p)$,

• $\mathcal{C}$-null if $v \in \partial C_p \cup \partial(-C_p)$,

• $\mathcal{C}$-timelike if $v \in C_p \cup (-C_p)$.
A *C*-null or *C*-timelike vector $v \in T_pM$ is said to be *C*-future directed if $v \in \overline{C_p}$ and *past directed if* $v \in (-\overline{C_p})$.

**Definition 2.3** (*C*-spacelike and *C*-null hyperplanes). Let $\mathcal{C} \subseteq \mathbb{V}^{d+1}$ be an open proper convex cone with vertex at the origin.

A linear hyperplane of $\mathbb{V}^{d+1}$ is said to be *C*-spacelike if it only intersect $\overline{\mathcal{C}}$ at its vertex 0 (see Figure 2.1).

A linear hyperplane of $\mathbb{V}^{d+1}$ is said to be *C*-null if it is a supporting hyperplane of *C* at point which is not the vertex of *C* (see Figure 2.1).

An affine hyperplane of $\mathbb{A}^{d+1}$ is said to be respectively *C*-null or *C*-spacelike if it is directed respectively by a *C*-null or a *C*-spacelike linear hyperplane.

The *future side* of a linear *C*-null or *C*-spacelike hyperplane is the open half-space it bounds which intersects *C*. Similarly, the *future side* of an affine *C*-null or *C*-spacelike affine hyperplane *H* is the open half-space it bounds which contains the future cone $p + \mathcal{C}$ of any point $p \in H$.

We denote respectively by $\mathcal{H}_{sp}(\mathcal{C})$ and $\mathcal{H}_n(\mathcal{C})$ the sets of $\mathcal{C}$-spacelike and $\mathcal{C}$-null linear hyperplanes of $\mathbb{V}^{d+1}$.

<figure><img src="image_3.png" /><figcaption>**Figure 2.1.** Illustrations of a *C*-spacelike hyperplane $H_1$, a *C*-null hyperplane $H_2$, and a hyperplane $H_3$ which is neither *C*-spacelike nor *C*-null.</figcaption></figure>

**Definition 2.4** (*C*-chronological and *C*-causal curves). Let $M$ be a smooth connected manifold endowed with a cone structure $\mathcal{C}$. A piecewise $\mathcal{C}^1$ curve in $M$ is *C*-chronological (respectively *C*-causal) if its velocity vectors are all *C*-timelike (respectively non-*C*-spacelike). In both cases we say that it is *future directed* (respectively *past directed*) if all its velocity vectors are future directed (respectively past directed).

**Definition 2.5** (*C*-future and *C*-past of a point, *C*-diamond). Let $p$ be a point of a smooth connected manifold $M$ endowed with a cone structure $\mathcal{C}$. The *C*-future of $p$ (respectively the *C*-past of $p$) is the subset $I^+(p)$ (respectively $I^-(p)$) of $M$ formed by the points which are end-points of future directed (respectively past directed) *C*-chronological curves starting at $p$.

Replacing the *C*-chronological curves by *C*-causal curves, we obtain the definition of the *C*-causal *future J*(p) (respectively *C*-causal *past J*−(p)) of *p*.

For two points $p, q \in M$ the (possibly empty) set $I^+(p) \cap I^-(q)$ is called a *C*-diamond, and the set $J^+(p) \cap J^-(q)$ is called a *C*-causal diamond.

**Definition 2.6** (Global hyperbolicity and Cauchy surface). A convex cone structure $\mathcal{C}$ on a smooth connected manifold $M$ is *globally hyperbolic* if there exists a $\mathcal{C}^1$ embedded hypersurface $S$ which is:

* *C*-spacelike, that is every tangent hyperplane $T_pS \subseteq T_pM$ is *C*-spacelike,

* a *Cauchy surface*, namely every inextensible *C*-causal curve intersects $S$ at exactly one point.

The following results for Lorentzian spacetimes are due to Geroch in [Ger70]. As their proofs only rely on topological and differential arguments, and not on the Lorentzian metric or shape of the cones in the tangent space, these results still hold for smooth manifolds with a proper convex cone structure [Min19].
**Proposition 2.7.** Let $M$ be a smooth connected manifold admitting a globally hyperbolic convex cone structure $C$ with a Cauchy surface $S$. All the Cauchy surfaces of $M$ for the cone structure $C$ are homeomorphic and $M$ is homeomorphic to $S \times \mathbb{R}$.

*Proof.* In order to be able to replicate the proof in [Ger70, Property 7], one just needs to produce a non-vanishing $C$-causal vector field on $M$. For that fix any Riemannian metric $g$ on $M$ and take $X(p) \in T_pM$ to be the barycentre with respect to $\text{vol}_{g_p}$ of the intersection $C_p \cap B_{g_p}$, where $B_{g_p}$ is the unit ball of $(T_pM, g_p)$. The vector field $X$ is $C$-causal because of the convexity of the cone distribution $C$, and non-vanishing because of its properness. Let $\phi_X^t$ be the flow associated with $X$, up to rescaling $X$ we can assume that it is always defined for all $t \in \mathbb{R}$. Every curve $(\phi_X^t(p))_{t \in \mathbb{R}}$ is an inextensible $C$-causal curve and thus intersects $S$. Hence the wanted homeomorphism is provided by

$$ \Phi: \begin{cases} S \times \mathbb{R} & \rightarrow M \\ (p, t) & \mapsto \phi_X^t(p) \end{cases} \qquad \square $$

**Theorem 2.8** ([Min19, Theorem 2.45]). A smooth connected manifold $M$ with a convex cone structure $C$ is globally hyperbolic if and only if it admits no closed causal curve and all the $C$-causal diamonds are compact.

2.3. **Affine spacetimes.** Given any open proper convex cone $C$ in $\mathbb{A}^{d+1}$, the affine space of dimension $d+1$ can be endowed with the parallel convex cone structure induced by $C$. Namely, taking $X_0$ to be the vertex of $C$, we can see $C_{X_0} = C$ as an open proper convex cone in $\mathbb{V}^{d+1} = T_{X_0}\mathbb{A}^{d+1}$ with vertex at the origin. Then any cone $C_X$ is defined as the parallel transports of $C_{X_0}$ from $T_{X_0}\mathbb{A}^{d+1}$ to $T_X\mathbb{A}^{d+1}$ by the flat affine connection $D$ on $\mathbb{A}^{d+1}$. We shall introduce affine spacetimes as manifolds with a cone structure which are locally modelled on $(\mathbb{A}^{d+1}, C)$.

**Definition 2.9** (Affine spacetime). An *affine spacetime* $(M, \mathcal{C})$ is a connected equiaffine manifold $M$ endowed with a convex cone structure $\mathcal{C}$ parallel for the flat affine connection induced by the equiaffine structure of $M$.

**Definition 2.10** (Maximality of a globally hyperbolic affine spacetime). A globally hyperbolic affine spacetime $(M, \mathcal{C})$ is said to be

* Cauchy-compact globally hyperbolic if it admits a compact Cauchy surface,

* maximal if, for every globally hyperbolic affine spacetime $(M', \mathcal{C}')$ and any embedding $i: M \hookrightarrow M'$ preserving the equiaffine structure, the cone structure, and mapping any Cauchy surface of $M$ to a Cauchy surface of $M'$, $i$ is necessarily surjective.

**Definition 2.11** (Convexity of a Cauchy surface of an affine spacetime). A Cauchy surface $S$ of a globally hyperbolic affine spacetime $(M, \mathcal{C})$ is said to be *convex* if in any local affine chart, the $\mathcal{C}$-spacelike surface $S$ can locally be parametrised as a the graph of convex function, and any $\mathcal{C}$-causal vector on $S$ points toward the convex side.

It is said to be locally uniformly convex if moreover those graphs are graphs of $C^2$ functions with positive definite Hessian matrix.

**Definition 2.12** (Isomorphism of affine spacetimes). An *isomorphism of affine spacetimes* between two $(d+1)$-dimensional affine spacetimes $(M, \mathcal{C})$ and $(M', \mathcal{C}')$ is a diffeomorphism

$$ \varphi: M \rightarrow M' $$

such that

* $\varphi$ preserves the equiaffine structure, i.e. on each chart of the equiaffine structure of $M$ the corestriction of $\varphi$ to a chart of the equiaffine structure of $M'$ is the restriction of an element of $\text{SA}(\mathbb{A}^{d+1})$,
• $\varphi$ sends the cone distribution $\mathcal{C}$ to $\mathcal{C}'$, i.e. for all $x$ in $M$,

$$d_x\varphi(\mathcal{C}_x) = \mathcal{C}'_{\varphi(x)}.$$

*Remark 2.13.* Imposing a parallel cone structure is a natural choice generalising the Lorentzian case, where the distribution of isotropic cones in the tangent bundle is parallel for the Levi-Civita connexion of the metric.

### 3. Backgrounds

Now, let us recall a few results from two classical theories which will prove useful for our work.

**3.1. Background on affine differential geometry and affine spheres.** This subsection is composed of a few results and definitions concerning *affine differential geometry* and *affine spheres*. For a more complete and deeper overview on those notions we refer the reader to [NS94, Lof10, LSZH15].

Affine differential geometry studies smooth hypersurfaces $\Sigma$ embedded in $\mathbb{A}^{d+1}$ through the following method. The datum of a transverse vector field $N: \Sigma \to \mathbb{V}^{d+1}$ allows one to decompose the tangent space at each point $p \in \Sigma$ as the direct sum

$$T_p\mathbb{A}^{d+1} = T_p\Sigma \oplus \langle N(p) \rangle.$$

Using that decomposition, one can define *affine invariants* of $(\Sigma, N)$:

• the *induced affine connection* $\nabla$ and the *induced affine fundamental form* or *Blaschke fundamental form* $h \in \Gamma(T^*\Sigma \otimes T^*\Sigma)$, determined by

$$D_X Y = \nabla_X Y + h(X, Y)N,$$

• the *affine shape operator* $S \in \Gamma(T^*\Sigma \otimes T\Sigma)$ and the *transversal connection form* $\tau \in \Gamma(T^*\Sigma)$, determined by

$$D_X N = S(X) + \tau(X)N,$$

• the *affine Gauss-Kronecker curvature* $\phi \in C^\infty(\Sigma, \mathbb{R})$, defined by

$$\phi = \det S,$$

• the *induced volume form* $\nu \in \Gamma(\bigwedge^{d+1} T^*\Sigma)$, defined by

$$\nu(X_1, \dots, X_d) = \det(X_1, \dots, X_d, N),$$

• in the case where the Blaschke fundamental form $h$ is non degenerate, the *volume form* $d\text{vol}_h \in \Gamma(\bigwedge^{d+1} T^*\Sigma)$ *induced by h* on $\Sigma$.

Given an open set $U \subset \mathbb{R}^d$, a smooth immersion $f: U \to \mathbb{A}^{d+1}$ and a smooth map $N: U \to \mathbb{V}^{d+1}$ transversal to $f$, the above invariants can all directly be defined on $U$ by pullback. In order to lighten them, all the following definitions and propositions will be stated for smooth hypersurfaces, but once again they can be extended to smoothly immersed hypersurfaces.

**Definition 3.1** (Equiaffine and normal affine fields). A transverse vector field $N$ on $\Sigma$ a smooth hypersurface in $\mathbb{A}^{d+1}$ is said to be *equiaffine* if the transversal connection form $\tau$ vanishes. An equiaffine vector field is said to be an *affine normal field* if moreover $\nu = d\text{vol}_h$.

**Proposition 3.2.** Let $\Sigma$ be a smooth hypersurface in $\mathbb{A}^{d+1}$. Here are some fundamental facts from the theory of affine differential geometry.

• The rank and signature of the affine fundamental form $h$ only depend on $\Sigma$ and not on the choice of the transverse vector field $N$. Thus $\Sigma$ is said to be non-degenerate if $h$ is, in that case $h$ is a genuine pseudo-Riemannian metric.
* If $\Sigma$ is locally convex, it is non-degenerate if and only if it is locally uniformly convex, meaning that $\Sigma$ can locally be parametrized as the graph of a map with positive definite Hessian matrix. Then $h$ is a Riemannian metric if and only if the transverse vector field $N$ is pointing towards the convex side.

* Any non-degenerate hypersurface admits a unique (up to sign) affine normal.

**Definition 3.3 (Affine conormal field).** Let $\Sigma$ be a smooth hypersurface in $A^{d+1}$ with an affine normal field $N$. Its affine conormal field $N^* : \Sigma \to (\mathbb{V}^{d+1})^*$ is the field defined in the following way: for all $p \in \Sigma$, $l = N^*(p)$ is the unique linear form such that

$$l(N(p)) = 1 \quad \text{and} \quad \ker l = T_p\Sigma.$$

**Definition 3.4 (Blaschke metric).** Let $\Sigma$ be a smooth locally uniformly convex hypersurface in $A^{d+1}$. The Riemannian metric $h$ given by the unique affine normal pointing towards the convex side of $\Sigma$ is called the *affine metric* or *Blaschke metric*.

**Definition 3.5 (Affine sphere).** A non-degenerate hypersurface $\Sigma$ is called an *affine sphere* if its shape operator with respect to an affine normal field is of the form $S = \lambda \mathrm{Id}$, where $\lambda$ is a real constant. It is called *proper* if $\lambda \neq 0$ and *improper* if $\lambda = 0$. If $\Sigma$ is locally convex, then it is called *hyperbolic* if for the affine normal pointing towards the convex side $\lambda > 0$ and *elliptic* if for that normal $\lambda < 0$.

It can be shown that being a proper affine sphere with shape operator $\lambda \mathrm{Id}$ is a condition equivalent to the existence of a point $o \in A^{d+1}$, called the *centre of the affine sphere*, such that $N(p) = \lambda \overline{op}$ is an affine normal field.

**Theorem 3.6 (Cheng-Yau [CY77, Gig81, CY86]).** For any open proper convex cone $C \subseteq A^{d+1}$, there exists a unique complete hyperbolic affine sphere $\Sigma_C \subset C$ centred at the vertex of $C$, asymptotic to $\partial C$ with affine shape operator the identity.

Here, *proper* again means that the cone is non-empty and with closure containing no entire line. The hypersurface is *complete* in the sense that the Riemannian metric on it induced by the ambient Euclidean metric is complete<fnref n="1" />, and it is *asymptotic to $\partial C$* in the sense that the distance from $X \in \Sigma$ to $\partial C$ with respect to an ambient Euclidean metric tends to 0 as $X$ goes to infinity in $A^{d+1}$.

**Proposition 3.7 ([Gig81, Proposition 1]).** Let $\Sigma \subset \mathbb{R}^{d+1}$ be a proper affine sphere. The image of the conormal map $N^*$ (Definition 3.3) on $\Sigma$ is a proper affine sphere in $(\mathbb{V}^{d+1})^*$ (seen as an affine space) centred at the origin. We shall call it the affine sphere dual to $\Sigma$.

If $\Sigma$ is locally convex, then $N^*(\Sigma)$ is of the same type (elliptic or hyperbolic) as $\Sigma$.

## 3.2. Background on convex analysis and Legendre-Fenchel transform.
Here are a few useful notions and results from convex analysis, we refer the reader to the classic literature [Roc97] for a richer view of the basis of the theory.

**Definition 3.8 (Legendre-Fenchel transform).** The Legendre-Fenchel transform or Legendre-Fenchel conjugate of a lower semi-continuous function $s: \mathbb{R}^d \to \mathbb{R} \cup \{+\infty\}$ is the function $s^*: (\mathbb{R}^d)^* \to \mathbb{R} \cup \{+\infty\}$ defined by

$$s^*(l) = \sup_{x \in \mathbb{R}^d} (l(x) - s(x)).$$

Equivalently, using the usual scalar product $\cdot$ on $\mathbb{R}^d$ to identify $(\mathbb{R}^d)^*$ to $\mathbb{R}^d$, it is defined on $\mathbb{R}^d$ by

$$s^*(y) = \sup_{x \in \mathbb{R}^d} (x \cdot y - s(x)).$$

<footnote n="1">¹In the case of an hyperbolic affine sphere, it is equivalent to completeness of the Blaschke metric [LSZH15, Theorem 3.54], but in a more general setting it is important to keep in mind the differences between the distinct kinds of completeness. A whole discussion on that subject can be found in [LSZH15, Section 3.4].</footnote>
The Legendre–Fenchel transform is a standard and fundamental object in analysis. It can easily be shown that:

* it is order reversing, in the sense that if $s \le f$ then $f^* \le s^*$,

* if $c$ is a constant function $(s+c)^* = s^* - c$,

* the Legendre–Fenchel transform of a function is lower semi-continuous and convex,

* the biconjugate $s^{**}$ of a function is its convex hull, that is the pointwise largest lower semi-continuous convex function bounded above by $s$. Hence if $s$ is convex and lower semi-continuous then $s^{**} = s$, and for any function $s$ (not necessarily convex) there are no new higher conjugates after the biconjugate as $s^{***} = s^*$.

The domain of a convex lower semi-continuous function $s: \mathbb{R}^d \to \mathbb{R} \cup \{+\infty\}$ is the convex domain

$$ \mathrm{dom}(s) := \{x \in \mathbb{R}^d \mid s(x) < +\infty\}. $$

**Proposition 3.9.** If $s$ is a lower semi-continuous convex function on $\mathbb{R}^d$ with non-empty domain $\mathrm{dom}(s)$ included in the Euclidean ball $B_{\mathrm{Euc}}(0, r)$, then $s^*$ has finite values and is $r$-Lipschitz on the whole $\mathbb{R}^d$.

Thus, if $U$ is a bounded open convex subset of $\mathbb{R}^d$ one can define the conjugate of a continuous convex function $s: U \to \mathbb{R}$ by taking the conjugate of the unique lower semi-continuous function restricting to $s$ on $U$, taking value $+\infty$ on $\mathbb{R}^d \setminus \bar{U}$ and values in $\mathbb{R} \cup \{+\infty\}$ on $\partial U$. Then $s^*$ has finite values on all $(\mathbb{R}^d)^*$ and $(s^{**})|_U = s$.

**Proposition 3.10.** Let $U$ be a bounded open convex subset of $\mathbb{R}^d$. A sequence $s_n$ of convex functions $U \to \mathbb{R}$ uniformly converges to some convex function $s: U \to \mathbb{R}$ if and only if $f_n = s_n^* : \mathbb{R}^d \to \mathbb{R}$ uniformly converges to some convex function $f: \mathbb{R}^d \to \mathbb{R}$.

*Proof.* For $s$ a finite valued convex function on $U$ with conjugate $f$, as $(s^{**})|_U = s$ and $(s_n^{**})|_U = s_n$, using the order reversing property of the Legendre–Fenchel transform we get the equivalence

$$ s_n - \varepsilon \le s \le s_n + \varepsilon \iff f_n - \varepsilon \le f \le f_n + \varepsilon. \quad \square $$

**Definition 3.11 (Subgradient and subdifferential).** Let $s: \mathbb{R}^d \to \mathbb{R} \cup \{+\infty\}$ be a lower semi-continuous convex function and $x_0 \in \mathbb{R}^d$, a linear form $l \in (\mathbb{R}^d)^*$ is a subgradient of $s$ at $x_0$ if for all $x \in \mathbb{R}^d$

$$ s(x_0) + l(x - x_0) \le s(x). $$

From now on, we shall use the usual scalar product $\cdot$ on $\mathbb{R}^d$ to identify $(\mathbb{R}^d)^*$ to $\mathbb{R}^d$, a subgradient of $s$ at $x_0$ can then be seen as a vector $y \in \mathbb{R}^d$ such that for all $x \in \mathbb{R}^d$

$$ s(x_0) + (x - x_0) \cdot y \le s(x). $$

The set of all subgradients of $s$ at $x_0$ is a convex set of $\mathbb{R}^d$ called the *subdifferential* of $s$ at $x_0$ and is denoted by $\partial s(x_0)$. If $x_0$ is in the interior of $\mathrm{dom}(s)$ its subdifferential is non-empty.

One can easily see that in the most general case

$$ \partial s_1(x) + \partial s_2(x) \subseteq \partial(s_1 + s_2)(x). $$

In some cases, that even becomes an equality.

**Theorem 3.12 ([Roc97, Theorem 23.8]).** Let $s_1, s_2: \mathbb{R}^d \to \mathbb{R} \cup \{+\infty\}$ be lower semi-continuous convex functions such that $\mathrm{dom}(s_1) \cap \mathrm{dom}(s_2)$ has non-empty interior, then for all $x \in \mathbb{R}^d$,

$$ \partial s_1(x) + \partial s_2(x) = \partial(s_1 + s_2)(x) $$
**Proposition 3.13** (Fenchel Inequality). Let $s$ be a lower semi-continuous convex function $s$ on $\mathbb{R}^d$ and $x, y \in \mathbb{R}^d$. The following, known as the Fenchel inequality, holds:

$$ (3.1) \qquad x \cdot y \leq s(x) + s^*(y). $$

Moreover the following conditions are equivalent:

(i) the equality is achieved,

(ii) $y \in \partial s(x)$,

(iii) $x \in \partial s^*(y)$.

**Proposition 3.14** ([Roc97, Theorems 25.1 and 25.5]). Let $s: \mathbb{R}^d \to \mathbb{R} \cup \{+\infty\}$ be a lower semi-continuous convex function. It is differentiable at a point $x \in \text{dom}(s)$ if and only if its subdifferential $\partial s(x)$ is a singleton. In that case $\partial s(x) = \{\text{grad } s(x)\}$ where grad is the gradient for the usual scalar product $\cdot$ on $\mathbb{R}^d$ that we used for duality.

A lower semi-continuous convex function $s: \mathbb{R}^d \to \mathbb{R} \cup \{+\infty\}$ is $C^1$ on a non-empty open set $U \subseteq \text{dom}(s)$ if and only if at every point of $U$ its subdifferential is a singleton.

**Proposition 3.15.** Let $U$ be a bounded open subset of $\mathbb{R}^d$ and $s: U \to \mathbb{R}$ a convex function. The Legendre-Fenchel transform $f = s^*: \mathbb{R}^n \to \mathbb{R}$ is a $C^1$ function if and only if $s$ is strictly convex.

*Proof.* • Assume that $s$ is strictly convex. Let $y \in \mathbb{R}^d$, for all $x \in \partial f(y)$, using Proposition 3.13 we have

$$ s(x) = x \cdot y - f(y). $$

Hence, if $x, x' \in \partial f(y)$, for all $t \in [0, 1]$ we have $(1-t)x + tx' \in \partial f(y)$ and

$$ s((1-t)x + tx') = (1-t)x \cdot y + tx' \cdot y - f(y) = (1-t)s(x) + ts(x'), $$

but as $s$ is strictly convex that means that $x = x'$ and hence $\partial f(y) = \{x\}$ so $f$ is $C^1$ by Proposition 3.14.

• Assume that $s$ is not strictly convex. Let $x, x'$ be two distinct point of $U$ such that for all $t \in [0, 1]$

$$ s((1-t)x + tx') = (1-t)s(x) + ts(x'). $$

Let $y \in \partial s(\frac{x+x'}{2})$ be a subgradient of $s$ at $\frac{x+x'}{2}$. We claim that $y$ is then also a subgradient of $s$ at $x$ and $x'$, meaning by Proposition 3.13 that $\{x, x'\} \subset \partial s^*(y)$ and thus that $s^*(y)$ is not a singleton and $f = s^*$ is not $C^1$, by Proposition 3.14. Let us prove our claim, using the definition of a subgradient (Definition 3.11) we get that $y \in \partial s(\frac{x+x'}{2})$ satisfies

$$ s\left(\frac{x+x'}{2}\right) + \frac{x-x'}{2} \cdot y \leq s(x) \quad \text{and} \quad s\left(\frac{x'+x}{2}\right) + \frac{x'-x}{2} \cdot y \leq s(x'). $$

The sum of these inequalities is $2s(\frac{x+x'}{2}) \leq s(x) + s(x')$, which happens to be an equality, thus both those inequalities must be equalities, so that

$$ s\left(\frac{x+x'}{2}\right) + \frac{x-x'}{2} \cdot y = s(x) \quad \text{and} \quad s\left(\frac{x'+x}{2}\right) + \frac{x'-x}{2} \cdot y = s(x'). $$

Using Proposition 3.13 we have

$$ s(x) = s\left(\frac{x+x'}{2}\right) + \frac{x-x'}{2} \cdot y = s^*(y) + \frac{x+x'}{2} \cdot y + \frac{x-x'}{2} \cdot y = s^*(y) + x \cdot y, $$

meaning that $y \in \partial s(x)$. Similarly we have $y \in \partial s(x')$. $\square$
## 4. $C$-CONVEX DOMAINS IN THE AFFINE SPACETIME $(\mathbb{A}^{d+1}, \mathcal{C})$

This section is devoted to generalisations of notions and tools from convex geometry in the Minkowski space to the model affine spacetimes $(\mathbb{A}^{d+1}, \mathcal{C})$ (introduced in subsection 2.3).

In this whole section, we let $\mathcal{C}$ be an open proper convex cone in an affine space of dimension $d+1$, that is a non-empty open convex cone with closure containing no entire line. We choose the vertex of the cone as our base point, so that our affine space is identified with $\mathbb{R}^{d+1}$ through a parametrisation such that

$$ (4.1) \qquad \mathcal{C} = \{t(x, 1) \mid x \in \Omega, t > 0\}, $$

where $\Omega$ is a bounded open convex domain of $\mathbb{R}^d$ containing 0. We are thus working inside the affine spacetime $(\mathbb{R}^{d+1}, \mathcal{C})$, where the cone distribution is given at any point $X \in \mathbb{R}^{d+1}$ by $\mathcal{C}_X = \mathcal{C} \subset \mathbb{R}^{d+1} = T_X \mathbb{R}^{d+1}$.

In an attempt to keep a coherent and consistent notation that allows the reader to easily identify in which spaces each element lies, we shall use the following notations:

* an element of $\mathbb{R}^d$ will be denoted by $x$ and an element of $\mathbb{R}^{d+1}$ by $X$,

* when using the usual scalar product on $\mathbb{R}^d$ and $\mathbb{R}^{d+1}$ for duality identifications, dual elements will be denoted by $y \in \mathbb{R}^d = (\mathbb{R}^d)^*$ and $Y \in \mathbb{R}^{d+1} = (\mathbb{R}^{d+1})^*$.

**4.1. $C$-convex and $C$-regular domains.** Following the work of Nie and Seppi [NS22], we generalise standard notions from Lorentzian Minkowski geometry to the affine spacetime $(\mathbb{R}^{d+1}, \mathcal{C})$.

**Definition 4.1** (Supporting hyperplane and Gauss map). Let $K$ a convex domain of $\mathbb{R}^{d+1}$ (it is thus open).

An affine hyperplane $H$ is said to be a *supporting hyperplane of K* if $H \cap K = \emptyset$ and there is a point $X \in H \cap \partial K$.

The Gauss map of $K$ is a set-valued map which maps each point $X$ of $\partial K$ to the set of linear hyperplanes directing a supporting hyperplane of $K$ at $X$.

Let $K$ be an open convex domain which is defined as an intersection of the future sides of $C$-spacelike hyperplanes of $\mathbb{R}^{d+1}$. Note that it may have $C$-null supporting hyperplanes (for instance, that is the case of the cone $\mathcal{C}$). We will denote by $\partial_{sp}K$ the part of the boundary $\partial K$ of $K$ which admits $C$-spacelike supporting hyperplanes (Definition 2.3).

**Definition 4.2** ($C$-convex and $C$-regular domains). A $C$-convex domain is a convex domain $K \subset \mathbb{R}^{d+1}$ which is an intersection of the future sides of $C$-spacelike hyperplanes of $\mathbb{R}^{d+1}$ and such that the restriction of its Gauss map to $\partial_{sp}K$ is surjective onto $\mathcal{H}_{sp}(\mathcal{C})$ (Definition 2.3).

A $C$-convex domain $K$ is called $C$-spacelike if all its supporting hyperplanes are spacelike, i.e. if it satisfies $\partial K = \partial_{sp}K$.

A $C$-convex domain $K$ is called $C^2$ if its spacelike boundary $\partial_{sp}K$ is a $C^2$ hypersurface and the restriction of its Gauss map to $\partial_{sp}K$ is a $C^1$ diffeomorphism (for the differentiable structure of the Grassmannian $\mathrm{Gr}_d(\mathbb{R}^{d+1})$, which is the space of linear hyperplanes of $\mathbb{R}^{d+1}$).

A $C$-regular domain is a $C$-convex domain which can be expressed as the intersection of the future sides of only $C$-null hyperplanes.

**Remark 4.3.** A trivial example of a $C$-regular domain is the cone $\mathcal{C}$ itself: it is the intersection of the future sides of all its $C$-null supporting hyperplanes. Its spacelike boundary is its vertex: $\partial_{sp}\mathcal{C} = \{0\}$.

**4.2. Support functions and affine sphere.** We shall now use the convex geometry tools developed in [FV16, BF17, BF20, NS22] in this more general setting. We set $\mathcal{C}^*$ to be the *polar cone* to $\mathcal{C}$ defined as the open proper convex cone

$$ \mathcal{C}^* := \mathrm{int} \left\{ l \in (\mathbb{R}^{d+1})^* \mid l(X) \le 0, \forall X \in \mathcal{C} \right\}. $$
Equivalently,

$$C^* = \{ l \in (\mathbb{R}^{d+1})^* \mid l(X) < 0, \forall X \in \bar{C} \setminus \{0\} \}.$$

For $K$ a $C$-convex domain of $\mathbb{R}^{d+1}$, we set its total support function $\tilde{s}_K$ (only denoted by $\tilde{s}$ when there is no possible confusion) to be the function

$$\tilde{s}_K : \begin{cases} C^* \to \mathbb{R} \\ l \mapsto \sup \{l(X) \mid X \in K\} \end{cases},$$

which is well-defined (the supremum is always finite) by definition of $C$-convex domains (Definition 4.2. It is also clearly sublinear and 1-homogeneous, hence convex.

From now on, using the usual scalar product $\cdot$ on $\mathbb{R}^{d+1}$, we identify $(\mathbb{R}^{d+1})^*$ to $\mathbb{R}^{d+1}$. Using identity (4.1), that allows us to see $C^*$ as $C^* = \{t(y, -1) \mid y \in \Omega^*, t > 0\}$ where $\Omega^*$ is the bounded open convex domain of $\mathbb{R}^d$ containing 0 defined by

$$\Omega^* = \{y \in \mathbb{R}^d \mid x \cdot y < 1, \forall x \in \bar{\Omega}\}.$$

We define $s_K : \Omega^* \to \mathbb{R}$, the $\Omega^*$-support function of a $C$-convex domain $K$ by $s_K(y) = \tilde{s}_K(y, -1)$. The total support function is 1-homogeneous so that for all $Y = (y, -\mu)$ in $C^*$,

$$\tilde{s}_K(Y) = \tilde{s}_K(y, -\mu) = \mu s_K\left(\frac{y}{\mu}\right).$$

As a total support function $\tilde{s}$ can fully be reconstructed from an $\Omega^*$-support function $s$, we shall mostly use the $\Omega^*$-support functions of $C$-convex domains. In order to lighten notations we shall simply write that $s$ is the support function of $K$.

**Proposition 4.4.** Let $K$ be a $C$-convex domain of $\mathbb{R}^{d+1}$ with support function $s$. The hypersurface $\partial K \subset \mathbb{R}^{d+1} = \mathbb{R}^d \times \mathbb{R}$ is the graph of $f = s^*$ the Legendre-Fenchel transform of $s$ (Definition 3.8).

$$\partial K = \text{graph}(f) := \{(x, f(x)) \mid x \in \mathbb{R}^d\}.$$

The domain $K$ is then its epigraph:

$$K = \text{epigraph}(f) := \{(x, \lambda) \in \mathbb{R}^d \times \mathbb{R} \mid \lambda > f(x)\}.$$

*Proof.* For $y \in \Omega^*$,

$$s(y) = \tilde{s}(y, -1) = \sup_{(x, \lambda) \in K} (x, \lambda) \cdot (y, -1) = \sup_{(x, \lambda) \in K} (x \cdot y - \lambda).$$

Since the supporting half-spaces of $K$ are $C$-spacelike and hence not vertical in our coordinates (as $0 \in \Omega^*$, we have $(0, \dots, 0, 1) \in C$), one can see $\partial K$ as a graph: $\partial K = \text{graph}(f)$ for some convex function $f: \mathbb{R}^d \to \mathbb{R}$. Hence the previous equation becomes

$$s(y) = \sup_{x \in \mathbb{R}^d} (x \cdot y - f(x)) = f^*(y),$$

and as $f$ is convex that means that $f = f^{**} = s^*$, $\partial K = \text{graph}(f) = \text{graph}(s^*)$, and $K = \text{epigraph}(f)$. $\square$

Conversely, it is clear that any convex function $s: \Omega^* \to \mathbb{R}$ gives a $C$-convex domain, denoted by $K(s)$, by taking the epigraph of its Legendre-Fenchel transform $f = s^*$, which is finite over the whole $\mathbb{R}^d$ by Proposition 3.9.

This tube domain $\Omega^* \times \mathbb{R}$, in which the graphs of support function live, generalises the setting of co-Minkowski geometry [FS19, BF20], also known as “half-pipe” geometry [Dan13]. The duality with that tube domain was already used and studied in the work of Choi [Cho17, Cho25], Nie and Seppi [NS22, NS23], or Bobb and Farre [BF24].
<figure><img src="image_4.png" /><figcaption>Figure 4.1. Convex duality in dimension $d+1=2$.</figcaption></figure>

Figure 4.1 illustrates that duality between $C$-convex domains and convex graphs in the tube domain $\Omega^* \times \mathbb{R}$.
It shows points

$$X_0 = (x_0, s^*(x_0)) \in \mathbb{R}^{d+1} \quad \text{and} \quad Y_0 = (y_0, s(y_0)) \in \Omega^* \times \mathbb{R},$$

with $x_0 \in \partial s(y_0)$. Proposition 3.13 implies that the hyperplanes

$$H_{Y_0} = \{(x, \lambda) \in \mathbb{R}^{d+1} \mid x \cdot y_0 - \lambda = s(y_0)\} \quad \text{and} \quad H_{X_0} = \{(y, \nu) \in \Omega^* \times \mathbb{R} \mid x_0 \cdot y - \nu = s^*(x_0)\},$$

are supporting hyperplanes of respectively graph(s*) and graph(s).

In that setting, Theorem 3.6 can now be more precisely stated.

**Theorem 4.5** (Cheng-Yau, [CY77, Gig81, CY86]). Let $C$ be open proper convex cone in the affine space $\mathbb{A}^{d+1}$, there is a unique complete hyperbolic affine sphere $\Sigma_C \subset C$ asymptotic to $\partial C$ with shape operator the identity. Parametrising $\mathbb{A}^{d+1}$ as $\mathbb{R}^{d+1}$ so that $C = \{t(x, 1) \mid x \in \Omega, t > 0\}$ with $\Omega$ a bounded convex domain containing 0, $\Sigma_C$ is the graph of the Legendre-Fenchel transform of the unique convex function $\omega_C$ satisfying the Monge-Ampère equation

$$\left\{ \begin{array}{l} \det \text{Hess } \omega = (-\omega)^{-d-2} \text{ in } \Omega^*, \\ \omega|_{\partial \Omega^*} = 0. \end{array} \right.$$

Moreover, $\omega_C$ satisfies $\lim_{y \to y_0} |\text{grad } \omega_C(y)| = +\infty$ for all $y_0 \in \partial \Omega^*$.

Then, $\Sigma_C^*$ the affine sphere polar to $\Sigma_C$ (the opposite of the dual affine sphere from Proposition 3.7) is the unique complete hyperbolic affine sphere with shape operator the identity and asymptotic to $\partial C^*$. It is given by the image of $-N_{\Sigma_C}^*$, the opposite of the conormal map on the affine sphere (Definition 3.3):

$$\Sigma_{(C^*)} = \Sigma_C^* = \left\{ -\frac{1}{\omega_C(y)}(y, -1) \mid y \in \Omega^* \right\}.$$

<figure><img src="image_5.png" /><figcaption>Figure 4.2. The affine sphere given by the theorem of Cheng and Yau.</figcaption></figure>
**Corollary 4.6.** Let $\mathcal{C}$ be open proper convex cone in the affine space $\mathbb{A}^{d+1}$. The affine sphere $\Sigma_{\mathcal{C}} = \text{graph}(\omega_{\mathcal{C}}^*)$ from Theorem 4.5 is preserved by $\text{Aut}_{\text{SA}}(\mathcal{C})$, the subgroup of elements of $\text{SA}(\mathbb{A}^{d+1})$ preserving the cone $\mathcal{C}$.

*Proof.* The action of an element $A \in \text{Aut}_{\text{SA}}(\mathcal{C})$ on $\mathbb{A}^{d+1}$ preserves the cone $\mathcal{C}$ and the volume, so it necessarily preserves the unique affine sphere $\Sigma_{\mathcal{C}}$ given by Theorem 4.5. $\square$

*Remark 4.7* (The Minkowski case). In the case where $\Omega$ is the Euclidean unit ball $\mathbb{B}^d$ in $\mathbb{R}^d$, then $\mathcal{C}$ is the Minkowski future light cone given by the inequalities $x_1^2 + \cdots + x_d^2 - x_{d+1}^2 < 0$ and $x_{d+1} > 0$, and $\Omega^* = \mathbb{B}^d$ is the unit ball given by $y_1^2 + \cdots + y_d^2 < 1$. The support function of the affine sphere is $\omega_{\mathcal{C}}(y) = -\sqrt{1 - y_1^2 - \cdots - y_d^2}$ and the affine sphere $\Sigma_{\mathcal{C}}$ is the hyperboloid $\mathbb{H}^d \subset \mathbb{R}^d$ given by the equation $x_1^2 + \cdots + x_d^2 - x_{d+1}^2 = -1$ and $x_{d+1} > 0$. The polar affine sphere $\Sigma_{\mathcal{C}}^*$ is also a hyperboloid given by the equation $y_1^2 + \cdots + y_d^2 - y_{d+1}^2 = -1$ and $y_{d+1} < 0$.

We define $\bar{s} : \Sigma_{\mathcal{C}}^* \to \mathbb{R}$, the $\Sigma_{\mathcal{C}}^*$-support function of a $\mathcal{C}$-convex domain $K$, by $\bar{s} = \tilde{s}|_{\Sigma_{\mathcal{C}}}$. Let us introduce the radial identification on the dual affine spheres:

$$R^* : \left\{ \begin{array}{rcl} \Sigma_{\mathcal{C}}^* & \to & \Omega^* \\ (y, \mu) & \mapsto & -\frac{y}{\mu} \end{array} \right. \quad \text{with inverse} \quad (R^*)^{-1} : \left\{ \begin{array}{rcl} \Omega^* & \to & \Sigma_{\mathcal{C}}^* \\ y & \mapsto & -\frac{1}{\omega_{\mathcal{C}}(y)}(y, -1) \end{array} \right. .$$

Notice that because $\tilde{s}$ is 1-homogeneous and because of the expression of $\Sigma_{\mathcal{C}}^*$ given by Theorem 4.5, it also satisfies $\bar{s} = -\frac{s}{\omega_{\mathcal{C}}} \circ R^*$. Thus the affine sphere $\Sigma_{\mathcal{C}}$ has the constant function $-1$ for $\Sigma_{\mathcal{C}}$-support function.

4.3. **Gauss map.** Notice that this duality setting gives a smooth diffeomorphism between $\Omega^*$ and $\mathcal{H}_{sp}(\mathcal{C})$, and a homeomorphism between $\partial\Omega^*$ and $\mathcal{H}_n(\mathcal{C})$; by $y \mapsto \{X \in \mathbb{R}^{d+1} | X \cdot (y, -1) = 0\}$.

Let $K$ be a $\mathcal{C}$-convex domain with support function $s$. Using those identifications, we now see the restriction to $\partial_{sp}K$ of the Gauss map as a set-valued map with values in $\Omega^*$. It can be expressed as

$$(4.2) \qquad \mathcal{G}_K : \left\{ \begin{array}{rcl} \partial_{sp}K & \to & \mathcal{B}(\Omega^*) \\ X & \mapsto & \partial s^*(\pi(X)) \end{array} \right. ,$$

where $\mathcal{B}(\Omega^*)$ is the Borel algebra of $\Omega^*$, and $\pi$ is the projection $\pi : \left\{ \begin{array}{rcl} \mathbb{R}^{d+1} & \to & \mathbb{R}^d \\ X = (x, \lambda) & \mapsto & x \end{array} \right.$.

If $s$ is strictly convex, and thus $s^* = C^1$ by Proposition 3.15, one can even express it as a true map

$$\mathcal{G}_K : \left\{ \begin{array}{rcl} \partial_{sp}K & \to & \Omega^* \\ X & \mapsto & \text{grad } s^*(\pi(X)) \end{array} \right. .$$

Using the Fenchel inequality (Proposition 3.13), we get that $\mathcal{G}_K$ is invertible if and only if $s$ has a gradient at each point of $\Omega^*$, in that case $s$ is $C^1$ (Proposition 3.14) and the inverse of the Gauss map is expressed as

$$(4.3) \qquad \mathcal{G}_K^{-1} : \left\{ \begin{array}{rcl} \Omega^* & \to & \partial_{sp}K \\ y & \mapsto & (\text{grad } s(y), \text{grad } s(y) \cdot y - s(y)) \end{array} \right. ,$$

giving a parametrisation of $\partial_{sp}K$ by $\Omega^*$. Then the global inversion theorem gives the following proposition.

**Proposition 4.8.** A $\mathcal{C}$-convex domain is $C_+^2$ (Definition 4.2) if and only if its support function is $C^2$ with positive definite Hessian everywhere. We shall call such functions $C_+^2$ support functions.

*Remark 4.9.* Notice that the Gauss map of the affine sphere gives a favourite smooth diffeomorphism

$$\mathcal{G}_{\Sigma_{\mathcal{C}}} : \Sigma_{\mathcal{C}} \to \Omega^*.$$

Moreover the radial identifications in $\mathbb{R}^{d+1}$ and $(\mathbb{R}^{d+1})^*$ give two favourite smooth diffeomorphisms

$$R : \Sigma_{\mathcal{C}} \to \Omega \quad \text{and} \quad R^* : \Sigma_{\mathcal{C}}^* \to \Omega^*.$$
Thus the followings are all smoothly diffeomorphic

$$ \Omega \approx \Omega^* \approx \Sigma_C \approx \Sigma_C^*. $$

Using $N_{\Sigma_C}^*$ the affine conormal map on the affine sphere (Definition 3.3), everything can be summarised through the following commutative diagram.

<figure><img src="image_6.png" /></figure>

**4.4. Boundary functions, g-convex domains and Cauchy developments of C-spacelike surfaces.**
As every convex function $s: \Omega^* \to \mathbb{R}$, i.e. every support function of a C-convex domain, extends uniquely to a convex lower semi-continuous function $s: \bar{\Omega}^* \to \mathbb{R} \cup \{+\infty\}$ (using an abusive notation), let us introduce the following objects.

**Definition 4.10** ($\Omega^*$-boundary function and g-convex domain). A function $g: \partial\Omega^* \to \mathbb{R} \cup \{+\infty\}$ is an $\Omega^*$-boundary function if there exist a convex function $s: \Omega^* \to \mathbb{R}$ extending lower semi-continuously to $g$ on the boundary $\partial\Omega^*$. A convex domain $K$, with support function $s_K: \Omega^* \to \mathbb{R} \cup \{+\infty\}$ extending lower semi-continuously to $g$ is then called a *g-convex domain*.

**Remark 4.11.** A *g-convex domain* is not necessarily *C-convex*, as its support function might take infinite values inside the domain $\Omega^*$. Nevertheless, if $g$ is finitely valued, then every *g-convex domain* is a *C-convex domain*.

**Definition 4.12** (Convex envelope of an $\Omega^*$-boundary function). Let $g: \partial\Omega^* \to \mathbb{R} \cup \{+\infty\}$ be an $\Omega^*$-boundary function. The *convex envelope* of $g$ is the pointwise largest convex lower semi-continuous function $s_g: \bar{\Omega}^* \to \mathbb{R} \cup \{+\infty\}$ with boundary values on $\partial\Omega^*$ given by $g$. It can be expressed in the following way: for all $y \in \bar{\Omega}^*$,

$$ s_g(y) = \sup \{a(y) \mid a \text{ is affine and } a \le g \text{ on } \partial\Omega^*\}. $$

Because of the order reversing property of the Legendre–Fenchel transform, the fact that the convex envelope of an $\Omega^*$-boundary function is the largest support function of a *g-convex domain* clearly implies the following.

**Proposition 4.13.** Let $g: \partial\Omega^* \to \mathbb{R} \cup \{+\infty\}$ be an $\Omega^*$-boundary function and consider its convex envelope $s_g: \bar{\Omega}^* \to \mathbb{R} \cup \{+\infty\}$.

* If $g$ (and thus $s_g$) is not constant and equal to $+\infty$, by Proposition 3.9, the Legendre–Fenchel transform $s_g^*: \mathbb{R}^d \to \mathbb{R}$ is finite. Then, the convex domain $D_g$ with support function $s_g$, i.e. the epigraph of $s_g^*$, is the maximal *g-convex domain* for inclusion, in the sense that every *g-convex domain* is included in $D_g$.

* If $g \equiv +\infty$, then the *g-convex* $D_g := \mathbb{R}^{d+1}$, i.e. the epigraph of $s_g^* \equiv -\infty$, is the maximal *g-convex domain* for inclusion.

**Remark 4.14.** Again, the maximal *g-convex domain* $D_g$ is not necessarily *C-convex*. Nevertheless, if $g$ is finitely valued, then $D_g$ is actually a *C-convex domain*.
**Proposition 4.15.** Let $g : \partial\Omega^* \to \mathbb{R} \cup \{+\infty\}$ be an $\Omega^*$-boundary function. The maximal $g$-convex domain $D_g$, given by Proposition 4.13, is $C$-regular (Definition 4.2). It is the intersection of the $C$-future of the family of $C$-null affine hyperplanes given by $g : \partial\Omega^* \to \mathbb{R} \cup \{+\infty\}$, that is

$$D_g = \bigcap_{y \in \partial\Omega^*} \{X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) < g(y)\}.$$

*Proof.* Let us set

$$K_g := \bigcap_{y \in \partial\Omega^*} \{X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) < g(y)\}.$$

If $X \in K_g$, its future $I^+(X) = X + C$ is a $C$-regular domain with the affine function $a_X(y) = X \cdot (y, -1)$ as $C$-support function. By definition of $K_g$, it satisfies $a_X < g$ on $\partial\Omega^*$. Conversely, any affine function $a$ on $\mathbb{R}^d$ can be expressed by $a(y) = X_a \cdot (y, -1)$ where $X_a \in \mathbb{R}^{d+1}$, and then $a < g$ on $\partial\Omega^*$ implies that $X_a \in K_g$. Thus, the support function of $K_g$ is the convex envelope of $g$ (Definition 4.12), meaning that $D_g = K_g$ is $C$-regular, with the desired expression. $\square$

Let us now highlight how maximal $g$-convex domains given by Proposition 4.13, which at first seem to be objects purely related to convex geometry, actually have nice causality properties in the affine spacetime $(\mathbb{R}^{d+1}, C)$.

**Definition 4.16** (Cauchy development). The Cauchy development of a $C$-spacelike $C^1$ hypersurface $S$ in $(\mathbb{R}^{d+1}, C)$ is the set of points $X$ of $\mathbb{R}^{d+1}$ such that all inextensible $C$-causal curves through $X$ meet $S$. We shall denote it by $D(S)$.

**Proposition 4.17.** Let $K$ be a $C$-spacelike $C$-convex domain (Definition 4.2) with $C^1$ boundary and support function $s_K : \Omega \to \mathbb{R}$ extending lower semi-continuously to an $\Omega^*$-boundary function $g : \partial\Omega^* \to \mathbb{R} \cup \{+\infty\}$. Its boundary $\partial K = \partial_{sp}K$ has Cauchy development $D(\partial K)$ equal to the maximal $g$-convex domain $D_g$.

*Proof.* By hypothesis, the boundary $\partial K = \partial_{sp}K$ is the entire graph of the convex function $s_K^* : \mathbb{R}^d \to \mathbb{R}$.

• First let us show that $D(\partial K) \subseteq D_g$. Let $X = (x, \lambda) \in D(\partial K)$, its future $I^+(X) = X + C$ is the $C$-regular domain having the affine function $a_X(y) = x \cdot y - \lambda$ as its support function. As $X$ is in the Cauchy development of $\partial K$, every $C$-null line through $X$ intersects $\partial K$. Thus every $C$-null hyperplane through $X$ intersects $\partial K$.

As $K$ is $g$-convex, it is contained in the maximal $g$-convex domain $D_g$, and thus every $C$-null hyperplane through $X$ also intersects $D_g$. By Proposition 4.15, $D_g$ is the following intersection of $C$-future sides of $C$-null hyperplane,

$$D_g = \bigcap_{y \in \partial\Omega^*} \{X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) < g(y)\},$$

Thus, every $C$-null hyperplane through $X$ directed by a $y \in \partial\Omega^*$ is in the future of the following hyperplane:

$$\{X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) = g(y)\}.$$

Using the support function $a_X$, that means that for all $y \in \partial\Omega^*$, $a_X(y) < g(y)$ (see Figure 4.3).

As $a_X$ is affine, by definition of $s_g$, the convex envelope of $g$ (Definition 4.12), that implies that $a_X < s_g$ on $\Omega^*$, i.e. $X \in D_g$.

• Now, let us show the reverse inclusion $D_g \subseteq D(\partial K)$. Let $X \in D_g$, its support function $a_X$ satisfies that for all $y \in \partial\Omega^*$, $a_X(y) < g(y)$. As $s_K$, the support function of $K$, continuously extends to $g$ on $\partial\Omega^*$, we claim that implies that every $C$-causal line through $X$ intersects $\partial K$ (see Figure 4.3). Then, the graph $\partial K = \text{graph}(s_K^*)$ separates $J^+(X) \cup J^-(X) = (X + \bar{C}) \cup (X - \bar{C})$ in two distinct connected components, ensuring that every inextensible causal curves through $X$ meet $\partial K$. That is precisely $X \in D(\partial K)$.
Now, let us prove our claim. Let $L$ be a $C$-causal line through $X$, it is directed by a vector $V = (x', 1) \in \bar{\Omega} \times \{1\}$ and expressed as

$$L = \{X_t := X + tV \mid t \in \mathbb{R}\}.$$

The $C$-convex domain $K$ has a horizontal supporting hyperplane, thus for $t$ small enough we have that $X_t \notin K$. Hence, in order to prove that $L$ intersects $\partial K$, we just have to show that for $t$ large enough we have $X_t \in K$. As the convex $K$ can be expressed as

$$K = \text{epigraph}(s_K^*) = \bigcap_{y \in \Omega^*} \{X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) < s_K(y)\},$$

we shall prove that for $t$ large enough, we have that for all $y \in \Omega^*$,

$$X_t \cdot (y, -1) = a_X(y) + t(x' \cdot y - 1) < s_K(y).$$

Notice that because $s_K$ extends continuously to $g > a_X$ on $\partial \Omega^*$, the subset $A := \{y \in \Omega^* | a_X(y) \ge s_K(y)\}$ is a compact subset of the convex domain $\Omega^*$. If $A$ is empty we have $a_X < s_K$ on $\Omega^*$ and thus $X = X_0 \in K$. Otherwise, as $A$ is compact and non-empty, we can set $m := \min\{x' \cdot y - 1 \mid y \in A \subset \Omega^*\} < 0$ (by definition of $\Omega^*$) and $M := \min\{s_K(y) - a_X(y) \mid y \in A\} \in \mathbb{R}$. For $t > 0$ and such that $M + tm < 0$, we have that for all $y \in \Omega^*$, if $y \in A$, then

$$X_t \cdot (y, -1) = a_X(y) + t(x' \cdot y - 1) \le s_K(y) + M + tm < s_K(y),$$

otherwise $y \notin A$, and then

$$X_t \cdot (y, -1) = a_X(y) + t(x' \cdot y - 1) \le a_X(y) < s_K(y),$$

concluding the proof. □

<figure><img src="image_7.png" /><figcaption>**Figure 4.3.** Section view of situation in the proof of Proposition 5.18.</figcaption></figure>

## 5. AFFINE DEFORMATIONS OF DIVISIBLE CONVEX CONES

In the following section, we properly introduce and describe the groups $\Gamma_\tau$ from Theorem 1.1. The existence of $D_\tau$, the unique maximal $\Gamma_\tau$-invariant $C$-convex domain from that same theorem, follows from the work of Choi [Cho25] (see Remark 5.16). Still, we proceed to construct that domain $D_\tau$ following the same approach as Nie and Seppi [NS23], as details of this construction will later be useful in our study of the affine spacetime $(D_\tau/\Gamma_\tau, C)$.

### 5.1. Convex projective structures on manifolds and divisible convex sets.
A real projective structure, or simply projective structure, on a smooth manifold $M$ of dimension $d$, consists of an atlas with values in the $d$-dimensional projective space $\mathbb{P}(\mathbb{R}^{d+1})$, such that the transition maps are restrictions of projective
transformations. In the language of $(G, X)$-structures, a projective structure is a $(\mathrm{PGL}(\mathbb{R}^{d+1}), \mathbb{P}(\mathbb{R}^{d+1}))$-structure. A manifold is said to admit a *convex projective structure* if it can be realised as the quotient $U/G$ of a domain $U$, included and convex in some affine chart of $\mathbb{P}(\mathbb{R}^{d+1})$, by the properly discontinuous action of a discrete subgroup $G$ of $\mathrm{PGL}(\mathbb{R}^{d+1})$. If $U$ is a *proper convex domain*, that is if it is bounded in some affine chart, the projective structure is called *properly convex*.

Following the nomenclature introduced by Benoist [Ben04, Ben08], an open proper convex domain $U$ of $\mathbb{P}(\mathbb{R}^{d+1})$ is said to be *divisible* by a discrete group $G < \mathrm{PGL}(\mathbb{R}^{d+1})$ if $G$ acts properly discontinuously and cocompactly on $U$.

Throughout all this section, we let $\mathcal{C}$ be an open proper convex cone of $\mathbb{R}^{d+1}$ divisible by a discrete torsion-free subgroup $\Gamma < \mathrm{SL}(\mathbb{R}^{d+1})$. That is, $\Gamma$ preserves $\mathcal{C}$ and $\mathbb{P}(\mathcal{C})$ is divisible by $G = \varphi(\Gamma)$ which is the image of $\Gamma$ by the natural projection morphism

$$ \varphi : \mathrm{SL}(\mathbb{R}^{d+1}) \to \mathrm{PGL}(\mathbb{R}^{d+1}). $$

Notice that $\Gamma$ and $G = \varphi(\Gamma)$ are isomorphic. Indeed if $d$ is even, it is clear, because $\varphi$ is an isomorphism. When $d$ is odd, $\varphi$ is a $2:1$ surjective morphism with kernel $\ker(\varphi) = \{\pm\mathrm{Id}\}$, but if $\gamma \in \Gamma$ preserves $\mathcal{C}$ then $-\gamma$ does not, so $-\gamma \notin \Gamma$, meaning that the restriction $\varphi|_{\Gamma}$ is an isomorphism onto its image.

We choose a parametrisation such that $\mathcal{C} = \{t(x, 1) | x \in \Omega, t > 0\}$ where $\Omega$ is a bounded open convex domain of $\mathbb{R}^d$ containing 0. The subgroup $\Gamma$ acts on different spaces by different manners:

* $\Gamma$ acts on elements $X \in \mathbb{R}^{d+1}$ by $\gamma X$, that action preserves the cone $\mathcal{C}$ and the unique affine sphere $\Sigma_{\mathcal{C}}$ given by Theorem 4.5 (Corollary 4.6),

* $\Gamma$ acts projectively on $\Omega$, i.e. by identifying both $\Gamma$ with $\varphi(\Gamma) < \mathrm{PGL}(\mathbb{R}^{d+1})$ and $\Omega$ with the projective domain $\mathbb{P}(\mathcal{C})$. Using the affine parametrisation, the affine sphere and the radial identification, that action can be written as $\gamma X = R(\gamma R^{-1}(x))$, where $R$ is the radial identification between $\Sigma_{\mathcal{C}}$ and $\Omega$.

* $\Gamma$ acts dually on elements $Y \in \mathbb{R}^{d+1}$ by $\gamma \star Y := \gamma^{-\top}Y$, where $\gamma^{-\top}$ is the inverse of the transpose of $\gamma$. That action preserves the cone $\mathcal{C}^*$, and the unique affine sphere $\Sigma_{\mathcal{C}}^*$, moreover the conormal map (Definition 3.3) is equivariant in the sense that for all $X \in \Sigma_{\mathcal{C}}$ and $\gamma \in \Gamma$,

$$ N_{\Sigma_{\mathcal{C}}}^{*}(\gamma X) = \gamma \star N_{\Sigma_{\mathcal{C}}}^{*}(X). $$

* $\Gamma$ acts dually projectively on $\Omega^*$, that action being expressed as $\gamma \star y := R^*(\gamma \star (R^*)^{-1}(y))$, where $R^*$ is the radial identification between $\Sigma_{\mathcal{C}}^*$ and $\Omega^*$.

The divisibility of $\mathcal{C}$ by $\Gamma$ tells us that the manifold $\Omega/\Gamma$ is compact and admits a properly convex projective structure, as it is naturally smoothly diffeomorphic to $\mathbb{P}(\mathcal{C})/\Gamma$. The smooth diffeomorphisms introduced in Remark 4.9 are all $\Gamma$-equivariant by definition of the actions. Thus, we get the following.

**Proposition 5.1.** Let $\mathcal{C} = \{t(x, 1) | x \in \Omega, t > 0\}$ be an open proper convex cone, where $\Omega$ is a bounded open convex domain of $\mathbb{R}^d$ containing 0. If $\mathcal{C}$ is divisible by a discrete torsion-free subgroup $\Gamma < \mathrm{SL}(\mathbb{R}^{d+1})$ then its polar cone $\mathcal{C}^*$ is also divisible by $\Gamma$ (acting dually). Moreover $\Omega/\Gamma$, $\Omega^*/\Gamma$, $\Sigma_{\mathcal{C}}/\Gamma$ and $\Sigma_{\mathcal{C}}^*/\Gamma$ are all smoothly diffeomorphic and compact.

Finally, here is a useful lemma concerning divisible convex sets.

**Lemma 5.2.** Let $A$ and $B$ be two open proper convex subsets of $\mathbb{P}(\mathbb{V}^{d+1})$ such that $A \subseteq B$. If there exists a discrete subgroup $\Gamma < \mathrm{PGL}(\mathbb{V}^{d+1})$ preserving both $A$ and $B$ and such that either $A$ or $B$ is divisible by $\Gamma$, then $A = B$.

*Proof.* First notice that we can assume $A$, the smallest convex subset, to be the one divisible by $\Gamma$. Otherwise, if $B$ is divisible by $\Gamma$, the dual convex subsets verify the reverse inclusion $B^* \subseteq A^*$, are both preserved by the dual action of $\Gamma$, and, by Proposition 5.1, the smallest convex subset $B^*$ is divisible by $\Gamma$.
As A is divisible by $\Gamma$, we already know that the action of $\Gamma$ on A is properly discontinuous, so let us now prove that the action of $\Gamma$ on B also is. Identifying B with the Cheng-Yau affine sphere in the cone over B, we can endow B with a complete Riemannian metric $h_B$ invariant under $\Gamma$, namely the Blaschke metric of the affine sphere (Theorem 4.5). Then, $\Gamma$ is a discrete subgroup of the Lie group Isom($B, h_B$) of isometries of the complete Riemannian manifold $(B, h_B)$, so it acts properly discontinuously on B.

Now that we know that the actions of $\Gamma$ on A and B are both properly discontinuous, we can consider the Hausdorff quotient spaces $A/\Gamma$ and $B/\Gamma$, with universal covers $A \to A/\Gamma$ and $B \to B/\Gamma$. The inclusion $i: A \hookrightarrow B$ descends to a local homeomorphism $i: A/\Gamma \hookrightarrow B/\Gamma$. The divisibility of A implies that $A/\Gamma$ is compact, thus $i$ is a local homeomorphism on a compact topological space so it is a covering map. Its lift to universal covers $i: A \to B$ is then a covering map between simply connected spaces, so it must be a homeomorphism, implying that $A = B$. $\square$

**5.2. Affine deformation and cocycle.** We remind the reader that we have set $\Gamma < SL(\mathbb{R}^{d+1})$ to be a torsion-free subgroup dividing a proper convex cone $C \subset \mathbb{R}^{d+1}$.

An affine deformation of $\Gamma$ is a subgroup $\Gamma_\tau \subset SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ obtained by adding translation parts to elements of $\Gamma$. Thus, an affine deformation of $\Gamma$ is determined by a translation function $\tau: \Gamma \to \mathbb{R}^{d+1}$. The condition that $\Gamma_\tau$ is a subgroup forces $\tau$ to satisfy the following cocycle relation: for all $\alpha, \beta \in \Gamma$

$$ (5.1) \qquad \tau(\alpha\beta) = \tau(\alpha) + \alpha\tau(\beta). $$

Two affine deformations $\Gamma_\tau$ and $\Gamma'_\tau$ are said to be equivalent if there exists $X \in \mathbb{R}^{d+1}$ such that for all $\alpha \in \Gamma$, $(\alpha, \tau(\alpha))$ is conjugate to $(\alpha, \tau'(\alpha))$ through the translation by $X$. That is, if there exists $X \in \mathbb{R}^{d+1}$ such that $\tau - \tau'$ satisfies that for all $\alpha \in \Gamma$,

$$ (5.2) \qquad (\tau - \tau')(\alpha) = \tau(\alpha) - \tau'(\alpha) = (I - \alpha)X, $$

where $I$ is the identity in $SL(\mathbb{R}^{d+1})$.

Equation (5.1) characterises the vector space of cocycles of $\Gamma$ in $\mathbb{R}^{d+1}$

$$ Z^1(\Gamma, \mathbb{R}^{d+1}) := \{\tau: \Gamma \to \mathbb{R}^{d+1} \mid \forall \alpha, \beta \in \Gamma, \tau(\alpha\beta) = \tau(\alpha) + \alpha\tau(\beta)\}, $$

while equation (5.2) characterises the vector space of coboundaries of $\Gamma$ in $\mathbb{R}^{d+1}$

$$ B^1(\Gamma, \mathbb{R}^{d+1}) := \{\tau_X: \Gamma \to \mathbb{R}^{d+1} \mid X \in \mathbb{R}^{d+1}\}, \text{ where } \tau_X(\alpha) := (I - \alpha)X. $$

Therefore the space of equivalence classes of affine deformations of $\Gamma$ is described by the first group of the $\mathbb{R}^{d+1}$-valued group cohomology of $\Gamma$, that is the vector space

$$ H^1(\Gamma, \mathbb{R}^{d+1}) := Z^1(\Gamma, \mathbb{R}^{d+1})/B^1(\Gamma, \mathbb{R}^{d+1}). $$

**Remark 5.3.** The existence of non-trivial affine deformations of a group $\Gamma < SL(\mathbb{R}^{d+1})$ dividing an open proper convex cone is not obvious. In dimension $2+1$, Nie and Seppi [NS23] have proved that when $\Gamma$ is isomorphic to $\pi_1(S_g)$, the fundamental group of the closed connected surface of genus $g \ge 2$, $\dim H^1(\Gamma, \mathbb{R}^{d+1}) = 6g - 6$. In higher dimension, it is already well-known [BF20, Theorem 3.4] that closed hyperbolic manifolds admitting a totally geodesic hypersurface have holonomy group $\Gamma < SO_0(d, 1)$ with a non trivial deformation group $H^1(\Gamma, \mathbb{R}^{d+1})$. In Appendix A, we extend those examples and produce, in any dimension, groups dividing an open proper convex domain, which is not projectively equivalent to an ellipse (i.e. giving the Klein ball model of the hyperbolic space), and admitting non-trivial affine deformations.

**Remark 5.4.** Notice that a subgroup $G < SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ is not always an affine deformation of its linear part. A necessary and sufficient condition for that is that any two distinct elements of G have distinct linear
parts. Equivalently, that is the case if and only if $G$ does not contain any non-trivial translation, i.e. element of the form $(I, \tau)$, where $I$ is the identity in $SL(\mathbb{R}^{d+1})$ and $\tau \neq 0$.

**5.3. Equivariant convex domains.** Throughout the rest of this section we let $\tau \in \mathbb{Z}^1(\Gamma, \mathbb{R}^{d+1})$ be a cocycle of the torsion-free subgroup $\Gamma$ dividing the cone $C$, and $\Gamma_\tau$ the associated affine deformation. From now on, in order to lighten the notation, we shall denote $\tau_\gamma := \tau(\gamma)$.

**Definition 5.5** ($\tau$-convex domain and $\tau$-support functions). A $\tau$-convex domain is a $C$-convex domain $K \subset \mathbb{R}^{d+1}$ invariant under the action of $\Gamma_\tau$. The $\tau$-support function of a $\tau$-convex domain is its $\Omega^*$-support function as a $C$-convex domain.

**Proposition 5.6.** Let $K$ be a $C$-convex domain with total support function $\tilde{s}$. If $\sigma \in SL(\mathbb{R}^{d+1}) \rtimes \mathbb{R}^{d+1}$ has linear part $\gamma$ and translation part $\tau$, the total support function of $\sigma K$, that we denote by $\sigma \cdot \tilde{s}$, satisfies

$$ (\sigma \cdot \tilde{s})(Y) = \tilde{s}(\gamma^{-1} \star Y) + \tau \cdot Y. $$

Thus $K$ is $\tau$-convex if and only if $\tilde{s}$ is $\tau$-equivariant, in the sense that for all $Y \in C^*$ and $\gamma \in \Gamma$, it satisfies

$$ (5.3) \qquad \tilde{s}(Y) = \tilde{s}(\gamma^{-1} \star Y) + \tau_{\gamma} \cdot Y. $$

*Proof.* For all $Y \in C^*$ and $\sigma = (\gamma, \tau) \in SL(\mathbb{R}^{d+1}) \rtimes \mathbb{R}^{d+1}$,

$$ \begin{align*} (\sigma \cdot \tilde{s})(Y) &= \sup_{X \in \sigma K} X \cdot Y = \sup_{X \in K} \sigma X \cdot Y, \\ &= \sup_{X \in K} \gamma X \cdot Y + \tau \cdot Y, \\ &= \sup_{X \in K} X \cdot \gamma^{\top} Y + \tau \cdot Y, \\ &= \tilde{s}(\gamma^{-1} \star Y) + \tau \cdot Y. \end{align*} \quad \square $$

Using the radial identification $R^*$ (Remark 4.9) and the $\Gamma$-invariance of the affine sphere $\Sigma_C^*$, we get the following.

**Corollary 5.7.** Let $K$ be a $C$-convex domain with support function $s$. If $\sigma \in SL(\mathbb{R}^{d+1}) \rtimes \mathbb{R}^{d+1}$ has linear part $\gamma$ and translation part $\tau$, the $\Omega^*$-support function of $\sigma K$, that we denote by $\sigma \cdot s$, satisfies

$$ (\sigma \cdot s)(y) = \frac{\omega_C(y)}{\omega_C(\gamma^{-1} * y)} s(\gamma^{-1} * y) + \tau \cdot (y, -1). $$

Thus $K$ is $\tau$-convex if and only if its support function $s$ is $\tau$-equivariant, in the sense that for all $y \in \Omega^*$ and $\gamma \in \Gamma$, it satisfies

$$ (5.4) \qquad s(y) = \frac{\omega_C(y)}{\omega_C(\gamma^{-1} * y)} s(\gamma^{-1} * y) + \tau_{\gamma} \cdot (y, -1). $$

**Proposition 5.8.** Let $K$ be a $\tau$-convex domain. Its Gauss map is an equivariant set valued map: with respect to the action of $\Gamma_\tau$ on $\partial K$ and the action of $\Gamma$ on $\Omega^*$, i.e. for all $X \in \mathbb{R}^{d+1}$ and $\gamma \in \Gamma$, we have the set equality

$$ \mathcal{G}_K(\gamma X + \tau_\gamma) = \gamma * \mathcal{G}_K(X) := \{\gamma * y \mid y \in \mathcal{G}_K(X)\}. $$

If moreover $K$ has a strictly convex support function, its Gauss map is a true equivariant map: with respect to the action of $\Gamma_\tau$ on $\partial K$ and the action of $\Gamma$ on $\Omega^*$, i.e. for all $X \in \mathbb{R}^{d+1}$ and $\gamma \in \Gamma$,

$$ \mathcal{G}_K(\gamma X + \tau_\gamma) = \gamma * \mathcal{G}_K(X). $$

**Proposition 5.9.** Let $K$ be a $\tau$-convex domain with a strictly convex support function $s$. Its boundary $\partial K$ is $C^1$ and the action of $\Gamma_\tau$ on its spacelike boundary $\partial_{sp}K$ is free and properly discontinuous. If moreover $s$ is $C^1$, then $\partial_{sp}K/\Gamma_\tau$ is homeomorphic to $\Omega/\Gamma$.
**Proof.** By Proposition 3.15, the strict convexity of $s$ implies that its Legendre–Fenchel transform $s^*$ is $C^1$, hence $\partial K = \text{graph}(s^*)$ is $C^1$. As the Gauss map $\mathcal{G}_K : \partial_{sp}K \to \Omega^*$ is continuous, surjective by definition of a $C$-convex domain (Definition 4.2), and equivariant by Proposition 5.8, and the action of $\Gamma$ on $\Omega^*$ is free and properly discontinuous, then the action of $\Gamma_\tau$ on $\partial_{sp}K$ is necessarily free and properly discontinuous. If moreover $s$ is $C^1$, the Gauss map is an equivariant homeomorphism with inverse map given by expression (4.3). The Gauss map thus descends to a homeomorphism from $\partial_{sp}K/\Gamma_\tau$ to $\Omega^*/\Gamma \simeq \Omega/\Gamma$ (by Proposition 5.1). $\square$

**5.4. Extension of $\tau$-support functions to the boundary.** The following propositions can be found in [NS23], for the reader's convenience we do not only state these propositions but also explicitly present their proofs, in any dimension and with notation coherent with the rest of this article.

**Proposition 5.10 ([NS23, Lemma 5.1]).** There exists a smooth $\tau$-equivariant function on $\Omega^*$, i.e. a smooth function satisfying (5.4) on $\Omega^*$.

**Proof.** As $\Omega^*/\Gamma$ is compact and $\pi_\Gamma : \Omega^* \to \Omega^*/\Gamma$ is its universal cover, we can set $(U_i)_{0\le i\le n}$ to be a finite open cover of $\Omega^*/\Gamma$ such that for all $i$,

$$V_i := \pi_\Gamma^{-1}(U_i) = \bigsqcup_{\gamma \in \Gamma} U_{i,\gamma},$$

where the union is disjoint, each $\pi_\Gamma|_{U_{i,\gamma}} : U_{i,\gamma} \to U_i$ is a homeomorphism, and for all $\gamma, \gamma' \in \Gamma$, $U_{i,\gamma\gamma'} = \gamma * U_{i,\gamma'}$. We set $(\psi_i)_{0\le i\le n}$ to be a partition of unity on $\Omega^*/\Gamma$ subordinate to $(U_i)_i$. It lifts to a partition of unity $(\Psi_i)_i$ subordinate to $(V_i)_i$ defined by $\Psi_i = \psi_i \circ \pi_\Gamma$. Let $u \in C^\infty(\Omega^*/\Gamma)$. We define functions $v_i \in C^\infty(V_i)$ by

$$v_i|_{U_{i,\gamma}}(y) = \frac{\omega_C(y)}{\omega_C(\gamma^{-1} * y)} u(\pi_\Gamma(y)) + \tau_\gamma \cdot (y, -1).$$

Thus $v_i : V_i \to \mathbb{R}$ satisfies for all $y$ in $V_i$ and $\gamma \in \Gamma$,

$$v_i(y) = \frac{\omega_C(y)}{\omega_C(\gamma^{-1} * y)} v_i(\gamma * y) + \tau_\gamma \cdot (y, -1),$$

which is precisely the equivariance condition (5.4). That implies that the function $v := \sum_i v_i \Psi_i$ is $\tau$-equivariant. $\square$

**Proposition 5.11 ([NS23, Proposition 5.2]).** There exists a smooth $\tau$-support function (Definition 5.5) on $\Omega^*$ which extends to a continuous function on $\bar{\Omega}^*$.

**Proof.** Let $v$ be a smooth $\tau$-equivariant function on $\Omega^*$ given by Proposition 5.10, let $K$ be a precompact open subset of $\Omega^*$ sufficiently large so that $\Omega^* = \bigsqcup_{\gamma \in \Gamma} \gamma * K$. As $\omega_C$ is smooth and strictly convex we can set $T > 0$ such that

$$s_+ := v - T\omega_C \quad \text{and} \quad s_- := v + T\omega_C$$

are respectively locally uniformly concave (i.e. with everywhere negative definite Hessian) and locally uniformly convex (i.e. with everywhere positive definite Hessian) on $K$. As $\omega_C$ is $\Gamma$-equivariant, they are also $\tau$-equivariant, and we claim that implies that they are respectively concave and convex on the whole $\Omega^*$. Then that means that $s_+$ extends to $g_+$, an upper semi-continuous function with values in $\mathbb{R} \cup \{-\infty\}$ on $\partial\Omega^*$, and that $s_-$ extends to $g_-$, a lower semi-continuous function with values in $\mathbb{R} \cup \{+\infty\}$ on the boundary $\partial\Omega^*$. As $\omega_C$ extends to the zero function on $\partial\Omega^*$, we have that $g_- = g_+$ is a continuous function with finite values on $\partial\Omega^*$.

Now, let us prove our claim. We have that $\tilde{s}_-$, the 1-homogeneous extension of $s_-$ to $C^*$, has a semi-definite positive Hessian on $\tilde{K} := \{t(y, -1) | y \in K, t > 0\}$. As for all $Y \in \gamma \star \tilde{K}$ with $\gamma \in \Gamma$,

$$\tilde{s}_-(Y) = \tilde{s}(\gamma^{-1} * Y) + \tau_\gamma \cdot Y = \tilde{s}(\gamma^T Y) + \tau_\gamma \cdot Y,$$
we get that $\tilde{s}_-$ has a semi-definite positive Hessian on $\gamma \star \tilde{K}$. Hence, as $C^* = \bigcup_{\gamma \in \Gamma} \gamma \star \tilde{K}$, we see that $\tilde{s}_-$ is convex, and so is $s_-$. The exact same can be done to show the concavity of $s_+$. $\square$

**Proposition 5.12.** There exists a unique continuous function, denoted by $g_\tau$, on the boundary $\partial\Omega^*$ such that any $\tau$-support function (or more generally any $\tau$-equivariant continuous function on $\Omega^*$) continuously extends on $\overline{\Omega^*}$ to a function coinciding with $g_\tau$ on $\partial\Omega^*$, i.e. every $\tau$-convex domain is $g_\tau$-convex (Definition 4.10).

*Proof.* Let $s$ be a $\tau$-support function extending to $g$ on $\partial\Omega^*$ given by Proposition 5.11. Let $s'$ be any other $\tau$-support function. Then $\frac{s-s'}{\omega_C}$ is $\Gamma$-invariant, it is the lift of a continuous function on the compact quotient $\Omega^*/\Gamma$. Thus it reaches a minimum $m$ and a maximum $M$, and we have

$$m\omega_C \leq s - s' \leq M\omega_C.$$

As $\omega_C$ tends to zero on the boundary $\partial\Omega^*$ (Theorem 4.5), we get that $s'$ continuously extends on $\overline{\Omega^*}$ to a function coinciding with $g$ on $\partial\Omega^*$. $\square$

*Remark 5.13.* We have shown that every $\tau$-convex domain is $g_\tau$-convex, but the converse is not true, a $g_\tau$-convex has a priori no reason to be invariant under the action of $\Gamma_\tau$.

**5.5. Maximal $\tau$-convex domain and Cauchy developments of $\tau$-invariant $C$-spacelike surfaces.** In this subsection, we prove the existence of the maximal invariant domain from Theorem 1.1 and show that it is the Cauchy development of the boundary of any $C$-spacelike $\tau$-convex domains.

Using the boundary map $g_\tau$ given by Proposition 5.12, we can use the results from subsection 4.4. Let $s_\tau$ be the convex envelope of $g_\tau$ (Definition 4.12). Using that it can be expressed in the following way: for all $y \in \Omega^*$,

$$s_{\tau}(y) = \sup \{a(y) \mid a \text{ is affine and } a \le g_{\tau} \text{ on } \partial\Omega^{*}\},$$

one can easily check that $s_\tau$ is $\tau$-equivariant, i.e. satisfies (5.3), and thus that $D_\tau := K(s_\tau)$ is a $\tau$-convex domain. Proposition 4.13 and 4.15 give the following.

**Proposition 5.14.** The $C$-convex domain $D_\tau := K(s_\tau)$, with support function $s_\tau$, is the maximal $\tau$-convex domain for inclusion, in the sense that every $\tau$-convex domain is included in $D_\tau$.

<figure><img src="image_8.png" /><figcaption>**Figure 5.1.** The unique $\tau$-equivariant boundary function $g_\tau$, and its convex envelope $s_\tau$.</figcaption></figure>

**Proposition 5.15.** The maximal $\tau$-convex domain $D_\tau$ is $C$-regular (Definition 4.2). It is the intersection of the $C$-future of the family of $C$-null affine hyperplanes given by the boundary function $g_\tau : \partial\Omega^* \to \mathbb{R}$ associated with $\tau$, i.e.

$$D_{\tau} = \bigcap_{y \in \partial \Omega^{*}} \{ X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) < g_{\tau}(y) \}.$$
**Remark 5.16.** In the work of Choi [Cho25], the existence of $\tau$-convex domains is given by Theorem 4.3.1. Indeed, we are in a setting where the affine subgroup $\Gamma_{\tau} < GL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ has linear part in $SL(\mathbb{R}^{d+1})$ and thus automatically verifies Choi's "uniform middle eigenvalue condition". Letting $\Omega_{\mathbb{S}^{d+1}} \subset \mathbb{S}^{d+1}$ be the extension at infinity of $C$ in $\mathbb{S}^{d+1}$, the double cover of $P(\mathbb{R}^{d+1})$, Theorem 4.3.1. states the existence of a convex domain $K \subset \mathbb{R}^{d+1}$ such that $(\Gamma_{\tau}, K, \Omega_{\mathbb{S}^{d+1}})$ is a "properly convex triple" and $\Gamma_{\tau}$ acts "asymptotically nicely" on $K$. The definitions of a "properly convex triple" and an "asymptotically nice" affine action [Cho25, Section 4.1] imply that the convex domain $K$ verifies $K+C = K$, that it has supporting hyperplanes directed by every elements of $\Omega^*$ and that it is $\Gamma_{\tau}$-invariant, i.e. that $K$ is $\tau$-convex.

The existence and uniqueness of the maximal $\tau$-convex domain $D_{\tau}$ then comes from Theorem 4.3.7 in [Cho25]. It states that there is a unique family family $(H_y)_{y \in \partial \Omega^*}$ of affine $C$-null hyperplane (given by the boundary function $g_{\tau}$ in our approach) such that for every properly convex affine triple $(\Gamma_{\tau}, K, \Omega_{\mathbb{S}^{d+1}})$, the domain $K$ has all $(H_y)_{y \in \partial \Omega^*}$ as supporting hyperplanes "at infinity". Then taking the intersection of the $C$-future of that family of $C$-null affine hyperplanes gives a unique maximal $\tau$-convex domain $D_{\tau}$ (Proposition 5.15).

**Definition 5.17 (Initial singularity).** The $C$-spacelike boundary $\partial_{sp}D_{\tau}$ of the maximal $\tau$-convex domain is called the initial singularity of $D_{\tau}$.

Using Proposition 4.17, we also have the following results about the Cauchy development of the boundary of $C$-spacelike $\tau$-convex domains.

**Proposition 5.18.** Let $K$ be a $C$-spacelike $\tau$-convex domain (Definition 4.2) with $C^1$ boundary. Its boundary $\partial K = \partial_{sp}K$ has Cauchy development $D(\partial K)$ (Definition 4.16) equal to the maximal $\tau$-convex domain $D_{\tau}$.

**Corollary 5.19.** Let $K$ be a $C$-spacelike $C^2$ $\tau$-convex domain (Definition 4.2). Its boundary $\partial K = \partial_{sp}K$ is a $\Gamma_{\tau}$-invariant locally uniformly convex Cauchy surface of the affine spacetime $(D_{\tau}, C)$.

**5.6. The two maximal domains.** Notice that, up to now, both the group $\Gamma$ and the cone $C$ it divides were fixed, but if $\Gamma$ divides the cone $C$ it also divides its opposite $-C$. The two affine spacetime structures $(\mathbb{R}^{d+1}, C)$ and $(\mathbb{R}^{d+1}, -C)$ are linked: the notions of $C$-null and $C$-spacelike hyperplane are the same as the notions of $(-C)$-null and $(-C)$-spacelike hyperplane, but time is "reversed" as the notion of $C$-future coincides with the notion of $(-C)$-past. Thus, to $\Gamma_{\tau}$ an affine deformation of $\Gamma$ we can associate two domains $D_{\tau}^C$ and $D_{\tau}^{-C}$.

Notice that using the $C$-duality, $(-C)$-convex domains are given by concave support functions on $\Omega^*$. In Proposition 5.12 we have seen that both convex and concave $\tau$-equivariant functions on $\Omega^*$ have the same continuous extension $g_{\tau}$ on $\partial \Omega^*$. Thus the two domains come from considering:

* $s_{\tau}^{-} : \Omega^* \to \mathbb{R}$ (that we previously only denoted as $s_{\tau}$) the pointwise largest convex function on $\Omega^*$ with graph under the graph of $g_{\tau}$ on $\partial \Omega^*$, i.e.

$$s_{\tau}^{-}(y) = \sup \{a(y) \mid a \text{ is affine and } a \le g_{\tau} \text{ on } \partial \Omega^{*}\},$$

* $s_{\tau}^{+} : \Omega^* \to \mathbb{R}$ the pointwise smallest concave function on $\Omega^*$ with graph over the graph of $g_{\tau}$ on $\partial \Omega^*$, i.e.

$$s_{\tau}^{+}(y) = \inf \{a(y) \mid a \text{ is affine and } a \ge g_{\tau} \text{ on } \partial \Omega^{*}\}.$$

Then, the graphs of $s_{\tau}^{-}$ and $s_{\tau}^{+}$ bound $CH^{*}(\tau)$, the convex hull of $\text{graph}(g_{\tau})$ inside the dual space $(\mathbb{R}^{d+1})^{*}$:

$$CH^{*}(\tau) = \{(y, \mu) \in \mathbb{R}^{d+1} \mid y \in \Omega^{*}, s_{\tau}^{-}(y) \le \mu \le s_{\tau}^{+}(y)\}.$$

Consequently, by Proposition 5.15, the two domains $D_{\tau}^C$ and $D_{\tau}^{-C}$ can be expressed as, respectively, the intersection of the $C$-future, and the intersection of the $(-C)$-future (i.e. the $C$-past), of the same family of
C-null affine hyperplanes given by the boundary function $g_τ : \partial\Omega^* \to \mathbb{R}$ associated with $τ$. That implies that $D_τ^C$ and $D_τ^{-C}$ are disjoint. One can also easily check that $D_τ^C = -D_{-τ}^{-C}$.

## 6. COSMOLOGICAL TIME ON C-CONVEX DOMAINS

In this section, we introduce a very useful tool for our study of maximal $τ$-convex domains: cosmological time on $C$-convex domains. We also prove that it satisfies the regularity and concavity conditions from Theorem 1.3. All the tools from this section can be used in an affine spacetime $(\mathbb{A}^{d+1}, \mathcal{C})$, where $\mathcal{C}$ is an open proper convex cone (not necessarily divisible). Thus, we work in the same setting as in Section 4: the ambient affine spacetime is $(\mathbb{R}^{d+1}, \mathcal{C})$ where $\mathcal{C} = \{t(x, 1) | x \in \Omega, t > 0\}$ with $\Omega$ a bounded open convex domain of $\mathbb{R}^d$ containing 0.

**Definition 6.1 (Gauge support function).** Let $\mathcal{C} = \{t(x, 1) | x \in \Omega, t > 0\}$ be an open proper convex cone in $\mathbb{R}^{d+1}$. A gauge support function is a $\Omega^*$-support function (see subsection 4.2) $\omega : \Omega^* \to \mathbb{R}$ satisfying the following conditions:

(GS1) $\omega$ is $C_+^2$, i.e. it is $C^2$ with positive definite Hessian on the whole $\Omega^*$,

(GS2) $\omega$ extends continuously to $\bar{\Omega}^*$ and $\omega|_{\partial\Omega^*} = 0$,

(GS3) for all $y \in \partial\Omega^*$, we have $\lim_{y' \to y} |\operatorname{grad} \omega_C(y')| = +\infty$ and thus $\partial\omega(y) = \emptyset$.

For the rest of this section, we let $\omega : \Omega^* \to \mathbb{R}$ be a gauge support function.

Let $\Sigma = \partial K(\omega) = \operatorname{graph}(\omega^*)$ be the boundary of the $C$-convex with support function $\omega$. One can check that conditions (GS1) and (GS2) imply that $\Sigma$ is included in $\mathcal{C}$ and asymptotic to $\partial\mathcal{C}$. Moreover, condition (GS3) tells us that $\Sigma$ is $C$-spacelike, i.e. all its supporting hyperplanes are $C$-spacelike. As a consequence of conditions (GS1) and (GS2), $\omega < 0$ on $\Omega^*$.

A nice candidate for $\omega$ is $\omega_C$, the support function of $\Sigma_C$ the Cheng-Yau affine sphere given by Theorem 4.5. Because $\Sigma_C$ also has the nice property to be invariant by all the elements of $\operatorname{SL}(\mathbb{R}^{d+1})$ preserving $\mathcal{C}$ it will give rise to tools behaving well with quotients. Hence, we shall make this choice in subsection 6.3 and later in Section 7 for our study of the quotients $D_τ/\Gamma_τ$ through their cosmological time.

### 6.1. Normal projection, cosmological time and foliation of C-convex domains.

**Lemma 6.2.** Let $K$ be a $C$-convex domain with support function $s$. For all $t > 0$ the function $(s + tw)^*$ is $C^1$.

*Proof.* It suffices to apply Proposition 3.15, noticing that as $s$ is convex and $\omega$ is strictly convex, for all $t > 0$ the function $s + tw$ is strictly convex. $\square$

**Definition 6.3 (C-spacelike graph).** For $f : \mathbb{R}^d \to \mathbb{R}$ a convex function with graph having only $C$-spacelike or $C$-null supporting hyperplanes, we define the *C-spacelike part of its graph* to be the set

$$ \operatorname{graph}_{sp}(f) = \{(x, f(x)) \mid \partial f(x) \subseteq \Omega^*\}. $$

**Lemma 6.4.** Let $s$ a lower semi-continuous function on $\bar{\Omega}^*$ and $t > 0$. We have

$$ \operatorname{graph}_{sp}(s + t\omega)^* = \operatorname{graph}(s + t\omega)^*. $$

Thus, by Proposition 4.4 and Lemma 6.2, the $C$-convex domain $K(s + tw)$ with support function $s + tw$ has a $C^1$ boundary satisfying

$$ \partial_{sp} K(s + t\omega) = \partial K(s + t\omega). $$

*Proof.* It suffices to prove that for all $x \in \mathbb{R}^d$, $\partial(s + t\omega)^*(x) \subseteq \Omega^*$. Let us proceed by contradiction and assume that we can set $x \in \mathbb{R}^d$ and $y \in \partial(s + t\omega)^*(x) \cap \partial\Omega^*$. Then by Proposition 3.13, $x \in \partial(s + t\omega)(y)$.
Condition (GS3) from Definition 6.1 is satisfied by $\omega$, so $\partial(t\omega)(y) = \emptyset$. Thus, using Theorem 3.12, we get that

$$ \partial(s + t\omega)(y) = \partial s(y) + \partial(t\omega)(y) = \partial s(y) + \emptyset = \emptyset, $$

which contradicts the fact that $x \in \partial(s + t\omega)(y)$. □

**Lemma 6.5.** Let $K$ be a $C$-convex domain with support function $s$. The set $\{\partial K(s + t\omega) | t > 0\} = \{\text{graph}(s + t\omega)^* | t > 0\}$ gives a $C^1$ foliation of $K$ by $C$-spacelike hypersurfaces.

*Proof.* • First, let us note that the leaves are indeed $C^1$ and $C$-spacelike hypersurfaces because of Lemma 6.4.

• Now, let us prove that those leaves are disjoint. To see that those spacelike graphs are disjoint it suffices to notice that if $x \in \mathbb{R}^n$ and $y = \text{grad}(s + t\omega)^*(x) \in \Omega^*$ (that is well-defined because of Lemmas 6.2 and 6.4), then for all $0 < t < t'$, as $\omega(y) < 0$, by Proposition 3.13,

$$ \begin{aligned} (s + t\omega)^*(x) &= x \cdot y - (s + t\omega)(y), \\ &< x \cdot y - (s + t'\omega)(y), \\ &< \sup_{y' \in \Omega^*} (x \cdot y' - (s + t'\omega)(y')) = (s + t'\omega)^*(x). \end{aligned} $$

• Finally, let us show that those leaves foliate the whole $K$, that is $K = \bigcup_{t>0} \partial K(s + t\omega)$. The inclusion $\bigcup_{t>0} \partial K(s + t\omega) \subseteq K$ is pretty clear. Indeed, if $x \in \mathbb{R}^n$ and $y = \text{grad } s^*(x)$ then for $t > 0$, as $\omega(y) < 0$, using the same reasoning as in the previous point we get $s^*(x) < (s + t\omega)^*(x)$, and thus

$$ \partial K(s + t\omega) = \text{graph} (s + t\omega)^* \subset \text{epigraph}(s^*) = K. $$

Let us then prove the reverse inclusion. Let $X = (x, \lambda) \in K, t > 0$ and $y \in \partial s^*(x)$. By Proposition 3.13,

$$ (6.1) \qquad s^*(x) = x \cdot y - s(y). $$

Thus for $t \ge 0$, using the Fenchel inequality (3.1) and then equality (6.1),

$$ (s + t\omega)^*(x) \ge x \cdot y - s(y) - t\omega(y) = s^*(x) - t\omega(y). $$

Hence, as $\omega(y) < 0$, we have that $\lim_{t \to +\infty} (s + t\omega)^*(x) = +\infty$. Thus, as $(s + t\omega)^*(x)$ is continuous in $t$ (by Proposition 3.10) and $s^*(x) < \lambda$ (because $X = (x, \lambda) \in K$), by the intermediate value theorem there is some $t > 0$ such that $\lambda = (s + t\omega)^*(x)$, i.e. such that $X \in \text{graph}(s + t\omega)^* = \partial K(s + t\omega)$. Thus, $K \subseteq \bigcup_{t>0} \partial K(s + t\omega)$. □

**Lemma 6.6.** Let $K$ be a $C$-convex domain with support function $s$. For all $t > 0$,

$$ \partial K(s + t\omega) = \{ X + t \mathcal{G}_{\Sigma}^{-1}(y) \mid X \in \partial_{sp} K, y \in \mathcal{G}_K(X) \}, $$

where $\mathcal{G}_K$, the Gauss map of the $C$-convex domain $K$, is the set-valued map defined by (4.2), and, as $\omega$ is $C_+^2$, $\mathcal{G}_{\Sigma}^{-1}$, the inverse of the Gauss map of $\Sigma = \partial_{sp} K(\omega)$, is a $C^1$ diffeomorphism defined by (4.3).

*Proof.* Remember that for any support function $u$, its associated $C$-convex domain $K(u)$ is the intersection of the $C$-future of its supporting hyperplane so that

$$ (6.2) \qquad \overline{K(u)} = \{ X \in \mathbb{R}^{d+1} \mid X \cdot (y, -1) \le u(y), \forall y \in \Omega^* \}, $$

and hence that

$$ (6.3) \qquad \partial K_{sp}(u) = \{ X \in \overline{K(u)} \mid \exists y \in \Omega^*, X \cdot (y, -1) = u(y) \}. $$

• First, let us show that

$$ \partial K(s + t\omega) \supseteq \{ X + t \mathcal{G}_{\Sigma}^{-1}(y) \mid X \in \partial_{sp} K, y \in \mathcal{G}_K(X) \}. $$
Let $X \in \partial_{sp}K$ and $y \in \mathcal{G}_K(X)$. We have that $y$ directs a supporting hyperplane to $K$ at $X$, thus, using identity (6.2), for all $y' \in \Omega^*$

$$X \cdot (y', -1) \leq s(y')$$

with equality when $y' = y$. Similarly, we also have that $y$ directs the supporting hyperplane to $\Sigma$ at $\mathcal{G}_{\Sigma}^{-1}(y)$, thus, using identity (6.2), for all $y' \in \Omega^*$

$$\mathcal{G}_{\Sigma}^{-1}(y) \cdot (y', -1) \leq \omega(y')$$

with equality when $y' = y$. Thus, letting $X_t = X + t\mathcal{G}_{\Sigma}^{-1}(y)$, for all $y \in \Omega^*$

$$X_t \cdot (y', -1) = X \cdot (y', -1) + t\mathcal{G}_{\Sigma}^{-1}(y) \cdot (y', -1) \leq s(y') + t\omega(y') = (s + t\omega)(y'),$$

with equality when $y' = y$, meaning, using identities (6.2) and (6.3), that

$$X_t \in \partial_{sp}K(s + t\omega) = \{ X \in \overline{K(s + t\omega)} \mid \exists y \in \Omega^*, X \cdot (y, -1) = (s + t\omega)(y) \}.$$

Thus, using Lemma 6.4,

$$\{X + t\mathcal{G}_{\Sigma}^{-1}(y) \mid X \in \partial_{sp}K, y \in \mathcal{G}_K(X)\} \subseteq \partial K_{sp}(s + t\omega) = \partial K(s + t\omega).$$

• Now, let us show that

$$\partial K(s + t\omega) \subseteq \{X + t\mathcal{G}_{\Sigma}^{-1}(y) \mid X \in \partial_{sp}K, y \in \mathcal{G}_K(X)\}.$$

Let $X_t = (x, (s + t\omega)^*(x)) \in \partial K(s + t\omega)$. Set $y = \text{grad}(s + t\omega)^*(x) \in \Omega^*$ and $X_0 = X_t - t\mathcal{G}_{\Sigma}^{-1}(y)$. Then, by Proposition 3.13, $x$ is a subgradient of $s + t\omega$ at $y$ and

$$(s + t\omega)^*(x) = x \cdot y - (s + t\omega)(y).$$

Thus, for all $y' \in \Omega^*$, using the expression $X_t = (x, (s + t\omega)^*(x))$ to develop, we get

$$(6.4) \qquad X_t \cdot (y', -1) = x \cdot y' - (s + t\omega)^*(x) = x \cdot (y' - y) + (s + t\omega)(y).$$

Moreover, using expression (4.3) for the Gauss map, for all $y' \in \Omega^*$,

$$(6.5) \qquad \mathcal{G}_{\Sigma}^{-1}(y) \cdot (y', -1) = \text{grad} \omega(y) \cdot (y' - y) + \omega(y).$$

Thus, for all $y' \in \Omega^*$, by computing (6.4) – $t \cdot (6.5)$, we get

$$X_0 \cdot (y', -1) = X_t \cdot (y', -1) - t\mathcal{G}_{\Sigma}^{-1}(y) \cdot (y', -1) = s(y) + (x - t \text{ grad } \omega(y)) \cdot (y' - y).$$

As $x \in \partial(s + t\omega)(y)$ and $\partial\omega(y) = \{\text{grad } \omega(y)\}$, Theorem 3.12 implies that $x - t \text{ grad } \omega(y)$ is a subgradient of $s$ at $y$. By definition of the subgradient, for all $y' \in \Omega^*$,

$$X_0 \cdot (y', -1) = s(y) + (x - t \text{ grad } \omega(y)) \cdot (y' - y) \leq s(y')$$

with equality when $y' = y$. Using identity (6.3), that means that $X_0 \in \partial_{sp}K$, and thus that

$$\partial K(s + t\omega) \subseteq \{X + t\mathcal{G}_{\Sigma}^{-1}(y) \mid X \in \partial_{sp}K, y \in \mathcal{G}_K(X)\}.$$

□

Lemmas 6.5 and 6.6 together give the following.

**Proposition 6.7.** Let $K$ be a $C$-convex domain with support function $s$. Any point $X \in K$ is uniquely decomposed as

$$X = P + t\mathcal{G}_{\Sigma}^{-1}(y),$$

where $t > 0$, $P \in \partial_{sp}K$ and $y \in \mathcal{G}_K(X) \subseteq \Omega^*$ (see Figure 6.1). Moreover, we then have

$$y = \mathcal{G}_{s+t\omega}(X) = \text{grad}(s + t\omega)^*(\pi(X)).$$
<figure><img src="image_9.png" /><figcaption>**Figure 6.1.** Normal projection onto the spacelike boundary of a C-convex.</figcaption></figure>

**Definition 6.8** (Normal projection, cosmological time and projecting normal). Let $K$ be a $C$-convex domain. In the decomposition of some element $X \in K$ from Proposition 6.7,

• we shall denote $P \in \partial_{sp}K$ by $P_K(X)$ and call it the normal projection of $X$ onto $\partial_{sp}K$,

• we shall denote $t \in \mathbb{R}_+^*$ by $\mathcal{T}_K(X)$ and call it the cosmological time of $K$ at $X$,

• we shall denote $y \in \Omega^*$ by $y_K(X)$ and call it the projecting normal from $X$ to $\partial_{sp}K$.

**Proposition 6.9.** Let $K$ be a $C$-convex domain with support function $s$. The cosmological time $\mathcal{T}_K: K \to \mathbb{R}_+^*$, normal projection $P_K: K \to \partial_{sp}K$ and projecting normal $y_K: K \to \Omega^*$ are all continuous.

*Proof.* • First let us prove that the cosmological time is continuous. The closed epigraph of a continuous function $f: \mathbb{R}^d \to \mathbb{R}$ is the closed set

$$ \overline{\text{epigraph}}(f) := \{(x, \lambda) \in \mathbb{R}^d \times \mathbb{R} \mid \lambda \ge f(x)\}. $$

For all $I = ]a, b[ \subset \mathbb{R}_+^*$, as the functions $(s + a\omega)^*$ and $(s + b\omega)^*$ are continuous, we have that

$$ \mathcal{T}_K^{-1}(I) = \bigcup_{t \in I} \partial K(s + t\omega) = \overline{\text{epigraph}}(s + a\omega)^* \setminus \overline{\text{epigraph}}(s + b\omega)^* $$

is open, so $\mathcal{T}_K$ is continuous.

• Now let us prove that $y_K$ is continuous. Let $X_n$ be a sequence of points of $K$ converging to $X$. We have that $y_K(X_n) = \text{grad}(s + \mathcal{T}_K(X_n)\omega)^*(\pi(X))$. As $\mathcal{T}_K(X_n)$ converges to $\mathcal{T}_K(X)$, by continuity of $\mathcal{T}_K$, we have that $(s + \mathcal{T}_K(X_n)\omega)^*$ converges pointwise to $(s + \mathcal{T}_K(X)\omega)^*$ and thus using [Roc97, Theorem 24.5] we have that $y_K(X_n)$ converges to $y_K(X) = \text{grad}(s + \mathcal{T}_K(X)\omega)^*(\pi(X))$.

• Finally, as $P_K = \text{Id}_K - \mathcal{T}_K \times (\mathcal{G}_\Sigma^{-1} \circ y_K)$, the continuities of $\mathcal{T}_K$, $\mathcal{G}_\Sigma^{-1}$ and $y_K$ imply the continuity of $P_K$. $\square$

We shall denote the level sets of the cosmological time by

$$ S_K^t := \mathcal{T}_K^{-1}(t) = \partial K(s + t\omega) = \partial_{sp}K(s + t\omega) = \text{graph}(s + t\omega)^*. $$

**Proposition 6.10.** Let $K$ be a $C$-convex domain with support function $s$. The cosmological time $\mathcal{T}_K$ on $K$ is $C^1$ and its gradient at any point $X \in K$ is expressed as

$$ \text{grad} \mathcal{T}_K(X) = \frac{(y_K(X), -1)}{\omega(y_K(X))}, $$

where $y_K(X)$ is the projecting normal from $X$ to $\partial_{sp}K$.

*Proof.* Let $X \in K$. In order to lighten the notation, we denote by $P := P_K(X) \in \partial_{sp}K$ the normal projection of $X$ onto $\partial_{sp}K$, by $y := y_K(X)$ the projecting normal and $N = \mathcal{G}_\Sigma^{-1}(y)$. The cone $I^+(P) = P + C$, is foliated by its intersections with:
<figure><img src="image_10.png" /><figcaption>**Figure 6.2.** Foliation of a $C$-convex by level hypersurfaces of its cosmological time.</figcaption></figure>

- the level surfaces of the cosmological time $(S_K^t)_{t>0}$,

- the hyperplanes directed by $y$, i.e. $(P + t H_y)_{t>0}$, where $H_y = \{V \in C \mid V \cdot (y, -1) = \omega(y)\}$,

- the $C$-spacelike hypersufaces $(P + t \Sigma)_{t>0}$.

For any vector $V \in C$, we set $g(V) := \mathcal{T}_K(P+V)$, it is the unique $t > 0$ such that $P+V \in S_K^t$. Similarly we define $f(V)$ to be the unique $t > 0$ such that $V \in tH_y$, it can explicitly be expressed as

$$f: \begin{cases} C & \to \mathbb{R} \\ V & \mapsto \frac{V \cdot (y, -1)}{\omega(y)} \end{cases}$$

and we set $h(V)$ to be the unique $t > 0$ such that $V \in t\Sigma$.

Inspired by the proof of Bonsante [Bon05, Proposition 4.3] in the Lorentzian case, we claim that those satisfy the hypothesis of the following fact:

Let $U$ be a non-empty open subset of $\mathbb{R}^{d+1}$ and $f, g, h: U \to \mathbb{R}$ be maps such that:

- $f \le g \le h$,

- $f$ and $h$ are $C^1$,

- there exist a point $X_0 \in U$ such that $f(X_0) = g(X_0) = h(X_0)$ and $d_{X_0}f = d_{X_0}h$.

Then $g$ is differentiable at $X$ and $d_{X_0}f = d_{X_0}g = d_{X_0}h$.

Let us prove our claim.

- Notice that for all $t > 0$, $S_K^t = \text{graph}(s + t\omega)^*$ and $P + t\Sigma = \text{graph}(a_p + t\omega)^*$, where $a_p(y) = P \cdot (y, -1)$ is the support function of the future cone $I^+(P) = P + C$. Notice that because $P \in \partial K(s)$, we have $a_p \le s$ and thus, using the order reversing property of the Legendre-Fenchel transform, $(s + t\omega)^* \le (a_p + t\omega)^*$ are convex maps with graphs having the common supporting hyperplane $P + tH_y$ at $P + tN$ (see Figure 6.3). Hence, by definition of $f, g$ and $h$, we have $f \le g \le h$.

- Because of the smoothness of hyperplanes and affine spheres, the maps $f$ and $g$ are clearly 1-homogeneous smooth maps.

- Finally for all $t > 0$, $f(tN) = g(tN) = h(tN) = t$ and

$$(6.6) \qquad \text{grad } f(tN) = \text{grad } h(tN) = \frac{(y, -1)}{\omega(y)}.$$

Thus, for all $t > 0$, $g$ is differentiable at $tN$ with gradient expressed by equation (6.6). Hence as for all $V \in C$, $\mathcal{T}_K(P+V) = g(V)$ and $X = P + \mathcal{T}_K(X)N$, $\mathcal{T}_K$ is differentiable at $X$ and its gradient at any point $X \in K$ is expressed as

$$\text{grad } \mathcal{T}_K(X) = \frac{(y, -1)}{\omega(y)} = \frac{(y_K(X), -1)}{\omega(y_K(X))}.$$

Then the continuity of the projecting normal $y_K$ (Proposition 6.9) gives that the cosmological time $\mathcal{T}_K$ is $C^1$. $\square$
<figure><img src="image_11.png" /><figcaption>Figure 6.3. Situation in the proof of Proposition 6.10.</figcaption></figure>

**Proposition 6.11.** Let $K$ be a $C$-convex domain with support function $s$. Every level set of the cosmological time $T_K$ on $K$ is a convex Cauchy surface of the affine spacetime $(K, C)$.

*Proof.* Every level set {$T_K = t$} is the boundary of the $C$-spacelike domain $K(s + t\omega)$. Because of condition (GS2) from Definition 6.1, the lower semi-continuous convex extensions (maybe with value in $\mathbb{R} \cup \{+\infty\}$) of $s$ and $s + t\omega$ on $\bar{\Omega}^*$ are equal on $\partial\Omega^*$. Then, the proof that $\partial K(s + t\omega)$ is a Cauchy surface of $(K, C)$ the same as the one of Proposition 5.18. The convexity of the Cauchy surface $\partial K(s + t\omega)$, in the sense of Definition 2.11, is clear. □

## 6.2. Length of causal curves in the affine spacetime $(\mathbb{A}^{d+1}, C)$.
This subsection discusses the cosmological time through the framework of pseudo-Finsler geometry described in [PY19, Bur23] and based on the work of Busemann [Bus67]. In that context we proceed to give a geometric interpretation of cosmological time, justifying its name [AGH98].

In the model affine spacetime $(\mathbb{A}^{d+1}, C)$, using our fixed gauge support functions $\omega$ satisfying properties (GS1), (GS2) and (GS3), from Definition 6.1, one can define a pseudo-Finsler norm of a $C$-timelike vector as the gauge of $\Sigma$: a vector $V \in C \subset \mathbb{R}^{d+1} = T_X \mathbb{A}^{d+1}$ has norm $\lambda > 0$ if and only if $V \in \lambda\Sigma$. As $\Sigma$ is asymptotic to $C$, the norm of a $C$-null vector is set to 0, so that the norm is continuous on $\bar{C}$.

**Remark 6.12.** Here $K(\omega)$, the $C$-convex domain bounded by $\Sigma$, is an example of a Valinor convex domain introduced by Buro in order to introduce pseudo-Finsler metrics on manifolds [Bur23, Section 3.2].

Because $\Sigma = \text{graph}(\omega^*) \subseteq C$ is the graph of a strictly convex $C^1$ function asymptotic to $\partial C$, we can parametrise $\Sigma$ radially, i.e. set $w: \Omega \to \mathbb{R}$ to be the convex function such that

$$ \Sigma = \left\{ -\frac{1}{w(x)}(x, 1) \mid x \in \Omega \right\}. $$

Then a simple computation shows that the pseudo-Finsler norm of vectors $V = (v, \nu) \in C \subset \mathbb{R}^d \times \mathbb{R}$ is expressed as the positive 1-homogeneous function

$$ F(V) := -\nu w\left(\frac{v}{\nu}\right). $$

**Remark 6.13.** When using $\omega = \omega_C$ the support function of the Cheng-Yau affine sphere $\Sigma_C$ given by Theorem 4.5, notice that Remark 4.7 implies that in the Minkowski case it precisely coincides with the Lorentzian norm of a causal vector. In the general case, using Gigena's work [Gig81, Lof10], we have that the radial parametrisation of an hyperbolic affine sphere is

$$ \Sigma_C = \left\{ -\frac{1}{w_C(x)}(x, 1) \mid x \in \Omega \right\}, $$
where $w_C = \omega_{(C^*)}$ is the unique solution to the Monge–Ampère equation

$$ \begin{cases} \det \text{Hess } w = (-w)^{-d-2} & \text{in } \Omega, \\ w|_{\partial\Omega} = 0. \end{cases} $$

**Definition 6.14** (Length of a $C$-causal curve and $C$-distance). The length of a $C$-causal curve $\gamma : I \to \mathbb{R}^{d+1}$ is defined as

$$ l_F(\gamma) := \int_I F(\gamma'(t))dt \geq 0. $$

The homogeneity of the function $F$ ensures that it does not depend on the parametrisation of the path $\gamma$.

Given $X_0 \in \mathbb{R}^{d+1}$ and $X_1 \in J^+(X_0) = X_0 + C$, the $C$-distance between $X_0$ and $X_1$ is

$$ \rho(X_0, X_1) := \sup \{l_F(\gamma) \mid \gamma \text{ is a future directed } C\text{-causal curve from } X_0 \text{ to } X_1\}. $$

**Proposition 6.15.** Let $X_0 \in \mathbb{R}^{d+1}$ and $X_1 \in J^+(X_0) = X_0 + C$. The $C$-distance between $X_0$ and $X_1$ is attained for the straight line from $X_0$ to $X_1$, i.e.

$$ \rho(X_0, X_1) = F(X_1 - X_0). $$

*Proof.* Let $\gamma : [0, 1] \to \mathbb{R}^{d+1}$ be a future directed $C$-causal curve from $\gamma(0) = X_0$ to $\gamma(1) = X_1$. We decompose it as $\gamma(t) = (x(t), \lambda(t))$, then up to reparametrisation we can assume that $\lambda(t) = (1-t)\lambda_0 + t\lambda_1$. Then

$$ l_F(\gamma) = \int_0^1 F(\gamma'(t))dt = -\int_0^1 \lambda'(t)w\left(\frac{x'(t)}{\lambda'(t)}\right)dt = -(\lambda_1 - \lambda_0)\int_0^1 w\left(\frac{x'(t)}{\lambda_1 - \lambda_0}\right)dt. $$

As $w$ is convex, the Jensen inequality gives

$$ \int_0^1 w\left(\frac{x'(t)}{\lambda_1 - \lambda_0}\right)dt \geq w\left(\int_0^1 \frac{x'(t)}{\lambda_1 - \lambda_0}dt\right) = w\left(\frac{x_1 - x_0}{\lambda_1 - \lambda_0}\right). $$

Because the path is future directed, $\lambda_1 - \lambda_0 \geq 0$, and thus

$$ l_F(\gamma) \leq -(\lambda_1 - \lambda_0)w\left(\frac{x_1 - x_0}{\lambda_1 - \lambda_0}\right) = F(X_1 - X_0), $$

where $F(X_1 - X_0)$ the length of the straight line from $X_0$ to $X_1$. $\square$

**Proposition 6.16** (Time inequality). If $X_1, X_2, X_3 \in \mathbb{R}^{d+1}$ are causally ordered, i.e. such that $X_2 \in J^+(X_1)$ and $X_3 \in J^+(X_2) \subseteq J^+(X_3)$, we have the time inequality

$$ \rho(X_1, X_2) + \rho(X_2, X_3) \leq \rho(X_1, X_3). $$

*Proof.* Set $X_i = (x_i, \lambda_i)$, for all $i \in \{1, 2, 3\}$ As $w$ is convex we have

$$ \frac{\lambda_2 - \lambda_1}{\lambda_3 - \lambda_1}w\left(\frac{x_2 - x_1}{\lambda_2 - \lambda_1}\right) + \frac{\lambda_3 - \lambda_2}{\lambda_3 - \lambda_1}w\left(\frac{x_3 - x_2}{\lambda_3 - \lambda_2}\right) \geq w\left(\frac{x_3 - x_1}{\lambda_3 - \lambda_1}\right). $$

Multiplying by $-(\lambda_3 - \lambda_1) \leq 0$, we get the wanted result. $\square$

*Remark 6.17.* This time inequality is one of the axioms describing a *timelike space* in the sense of Busemann [Bus67]. One can easily check that a $C$-convex domains endowed with the causal structure induced by $C$ and the time function $\rho$ satisfies all the other axioms.

**Proposition 6.18.** Let $K$ a $C$-convex domain. The cosmological time defined in Definition 6.8 satisfies

$$ \begin{aligned} \mathcal{T}_K(X) &= \sup \{\rho(X_0, X) \mid X_0 \in J^-(X) \cap \partial_{sp} K\}, \\ &= \sup \{l_F(\gamma) \mid \gamma \text{ is a future directed } C\text{-causal curve from a point of } \partial_{sp} K \text{ to } X\}. \end{aligned} $$
Proof. First, notice that $T := \mathcal{T}_K(X) = F(X - P) = \rho(P, X)$ where $P := P_K(X) \in J^-(X) \cap \partial_{sp}K$ is the normal projection of $X$ onto $\partial_{sp}K$. Then notice that $X - T\Sigma$ and $\partial_{sp}K$ touch at $P$ and have a common $C$-spacelike hyperplane directed by $y := y_K(X)$ the projecting normal from $X$ to $\partial_{sp}K$. Then, using the foliation of $(X - C)$ by the $-C$-convex hypersurfaces $(X - t\Sigma)_{t>0}$, as $K$ is $C$-convex, it is clear that any element $X_0 \in (X - C) \cap \partial_{sp}K$ belongs to some $X - T_0 \cdot \Sigma$ with $T_0 \le \mathcal{T}_K(X)$, i.e. $\rho(X_0, X) \le \mathcal{T}_K(X)$ with equality for $X_0 = P$ (see Figure 6.4). $\square$

<figure><img src="image_12.png" /><figcaption>Figure 6.4. Situation in the proof of Proposition 6.18.</figcaption></figure>

Remark 6.19. This new characterisation of cosmological time highlights its similarity with the Lorentzian one [AGH98]. When using $\omega = \omega_C$, the support function of the Cheng-Yau affine sphere given by Theorem 4.5, as our gauge support function, Remark 4.7 implies that in the Minkowski case it precisely coincides with the Lorentzian cosmological time.

**Proposition 6.20.** Let $K$ be a $C$-convex domain. The cosmological time $\mathcal{T}_K$ on $K$ is concave.

Proof. Let $X_1, X_2 \in K$ be two points, we denote by $P_i := P_K(X_i) \in \partial_{sp}K$ their normal projections. Then we have for $i \in \{1, 2\}$,

$$ \mathcal{T}_K(X_i) = \rho(P_i, X_i) = F(X_i - P_i). $$

Starting from

$$ \begin{aligned} \frac{\mathcal{T}_K(X_1) + \mathcal{T}_K(X_2)}{2} &= \frac{F(X_1 - P_1) + F(X_2 - P_2)}{2}, \\ &= \frac{F(X_1 - P_1) + F((X_1 + X_2 - P_2) - X_1)}{2}, \\ &= \frac{\rho(P_1, X_1) + \rho(X_1, X_1 + X_2 - P_2)}{2}, \end{aligned} $$

and using the time inequality (Proposition 6.16),

$$ \rho(P_1, X_1) + \rho(X_1, X_1 + X_2 - P_2) \le \rho(P_1, X_1 + X_2 - P_2) = \rho(P_1 + P_2, X_1 + X_2), $$

we get

$$ \frac{\mathcal{T}_K(X_1) + \mathcal{T}_K(X_2)}{2} \le \frac{\rho(P_1 + P_2, X_1 + X_2)}{2} = \rho\left(\frac{P_1 + P_2}{2}, \frac{X_1 + X_2}{2}\right). $$

As $K$ is convex and $P_1, P_2 \in \partial_{sp}K$, we have that $\frac{1}{2}(P_1 + P_2) \in \bar{K}$ and thus, by Proposition 6.18,

$$ \frac{\mathcal{T}_K(X_1) + \mathcal{T}_K(X_2)}{2} \le \rho\left(\frac{P_1 + P_2}{2}, \frac{X_1 + X_2}{2}\right) \le \mathcal{T}_K\left(\frac{X_1 + X_2}{2}\right) \square $$

$\square$
**6.3. Invariance of the cosmological time given by the Cheng-Yau affine spheres.** In this subsection we use the support function $\omega_C$ of $\Sigma_C$, the Cheng-Yau affine sphere given by Theorem 4.5, as our gauge support function in order to prove Theorem 1.3.

Remember that the affine sphere $\Sigma_C = \text{graph}(\omega_C^*)$ is preserved by Aut$_{SL}(C)$, the subgroup of elements of $\text{SL}(\mathbb{R}^{d+1})$ preserving the cone $C$ (Corollary 4.6). That implies that the cosmological time induced by the Gauge support function $\omega_C$ satisfy the following invariance property.

**Proposition 6.21.** Let $K$ be a $C$-convex domain and $\mathcal{T}_K$, $P_K$ and $y_K$ be the cosmological time, normal projection and projecting normal on $K$ (Definition 6.8) induced by the gauge support function $\omega_C$, the support function of $\Sigma_C$ the Cheng-Yau affine sphere (Theorem 4.5). If $K$ is preserved by an affine transformation $(\gamma, \tau) \in \text{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$, then $\gamma \in \text{Aut}_{\text{SL}(C)}$, the subgroup of elements of $\text{SL}(\mathbb{R}^{d+1})$ preserving $C$, and for all $X$ in $K$ we have the following

$$ 
\begin{aligned}
\mathcal{T}_K(\gamma X + \tau) &= \mathcal{T}_K(X), \\
P_K(\gamma X + \tau) &= \gamma P_K(X) + \tau, \\
y_K(\gamma X + \tau) &= \gamma * y_K(X),
\end{aligned}
$$

where the projective dual action of an element $\gamma \in \text{Aut}_{\text{SL}(C)}$ on $\Omega^*$, denoted by $*$, is described at the beginning of Section 5.

*Proof.* Let $(\gamma, \tau) \in \text{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ be an affine transformation preserving the $C$-convex domain $K$. As $(\gamma, \tau)$ preserves $K$, its linear part $\gamma$ preserves its set of supporting hyperplane. By definition of a $C$-convex domain (Definition 4.2), that implies that the dual action of $\gamma$ preserves the cone $C^*$ and hence, by duality, that $\gamma$ preserves $C$, i.e. $\gamma \in \text{Aut}_{\text{SL}(C)}$.

Let $X \in K$. Acting by $(\gamma, \tau)$ on the equality

$$ X = P_K(X) + \mathcal{T}_K(X) \cdot \mathcal{G}_{\Sigma_C^{-1}}(y_K(X)) $$

that defines the cosmological time, normal projection and projecting normal of $X$ (Definition 6.8), we get

$$ \gamma X + \tau = (\gamma P_K(X) + \tau) + \mathcal{T}_K(X) \cdot \gamma \mathcal{G}_{\Sigma_C^{-1}}(y_K(X)). $$

As the affine sphere $\Sigma_C$ is invariant under the action of Aut$_{\text{SL}(C)}$ (Corollary 4.6), its Gauss map is clearly Aut$_{\text{SL}(C)}$-equivariant, and so is its inverse. Thus, using that equivariance of $\mathcal{G}_{\Sigma_C^{-1}}$, the previous equality becomes

$$ (6.7) \qquad \gamma X + \tau = (\gamma P_K(X) + \tau) + \mathcal{T}_K(X) \cdot \mathcal{G}_{\Sigma_C^{-1}}(\gamma * y_K(X)). $$

As $K$ is preserved by the affine transformation $(\gamma, \tau)$ and $\gamma \in \text{Aut}_{\text{SL}(C)}$, its $C$-spacelike boundary $\partial_{sp}K$ is also preserved by $(\gamma, \tau)$. Thus, $\gamma P_K(X) + \tau \in \partial_{sp}K$ and $\gamma * y_K(X)$ directs a supporting hyperplane to $K$ at that point. Hence, equality (6.7) characterises the cosmological time, normal projection and projecting normal of $\gamma X + \tau$ (Definition 6.8), giving the desired expressions. $\square$

We can now prove the following theorem, implying Theorem 1.3.

**Theorem 6.22.** Let $K$ be a $C$-convex domain. The cosmological time $\mathcal{T}_K: K \to \mathbb{R}_+^*$, induced by the gauge support function $\omega_C$, the support function of $\Sigma_C$ the Cheng-Yau affine sphere given by Theorem 4.5, is a $C^1$ function whose level sets are $C^1$ convex Cauchy surfaces of $(K, C)$ foliating $K$. Moreover, $\mathcal{T}_K$ is invariant by any transformation of $\text{SA}(\mathbb{R}^{d+1})$ preserving $K$.

*Proof.* This is just a consequence of Lemma 6.5 and Propositions 6.10, 6.11, 6.20 and 6.21. $\square$
## 7. QUOTIENT OF THE MAXIMAL DOMAIN

In this section, we first study the action of $\Gamma_\tau$, an affine deformation of a group $\Gamma$ dividing an open proper convex cone $C$, on the maximal $\tau$-convex domain $D_\tau$ from Theorem 1.1 constructed in Section 5. We then focus on the affine spacetime $(D_\tau/\Gamma_\tau, C)$, using the cosmological time we introduced in the previous section. That allows us to produce proofs for point (1) and the GHCC part of point (2) of Theorem 1.1.

We shall use the cosmological time functions introduced in Section 6 and given by setting $\omega = \omega_C$ the support function of the Cheng-Yau affine sphere given by Theorem 4.5.

### 7.1. Action of $\Gamma_\tau$ on $D_\tau$
In this subsection we shall prove point (1) from Theorem 1.1. We come back to the case where $C$ happens to be divisible by a torsion-free discrete subgroup $\Gamma < SL(\mathbb{R}^{d+1})$. Set $\Gamma_\tau$ to be some affine deformation of $\Gamma$, we can use the tools introduced in subsection 6.1 on the maximal $\tau$-convex domain $D_\tau$ constructed in Section 5.

In order to lighten notations, the normal projection onto the initial singularity $\partial_{sp}D_\tau$ will simply be denoted by $P_\tau$, similarly the cosmological time on $D_\tau$ will be denoted by $\mathcal{T}_\tau$, its $t$-level hypersurface by $S_\tau^t$, and the projecting normal by $y_\tau$.

**Proposition 7.1.** On $D_\tau$ the maximal $\tau$-convex domain:

* the cosmological time $\mathcal{T}_\tau: D_\tau \to \mathbb{R}_+^*$ is $\Gamma_\tau$-invariant,

* the normal projection $P_\tau: D_\tau \to \partial_{sp}D_\tau$ is $\Gamma_\tau$-equivariant,

* the projecting normal $y_\tau: D_\tau \to \Omega^*$ is equivariant with respect to the action of $\Gamma_\tau$ on $D_\tau$ and the action of $\Gamma$ on $\Omega^*$.

*Proof.* For all $X \in D_\tau$ and $(\gamma, \tau_\gamma) \in \Gamma_\tau$, by Proposition 6.21,

$$ \mathcal{T}_{\tau}(\gamma X + \tau_{\gamma}) = \mathcal{T}_{\tau}(X), $$

$$ P_{\tau}(\gamma X + \tau_{\gamma}) = \gamma P_{\tau}(X) + \tau_{\gamma}, $$

$$ y_{\tau}(\gamma X + \tau_{\gamma}) = \gamma * y_{\tau}(X). $$

□

**Proposition 7.2.** The maximal $\tau$-convex domain $D_\tau$ is foliated by the level hypersurfaces of its cosmological time, which are the spacelike $C^1$ embedded $\Gamma_\tau$-invariant hypersurfaces $\{\partial K(s_\tau + t\omega_C)\}_{t>0}$.

*Proof.* The foliation is simply a consequence of Lemma 6.5, the $\tau$-invariance comes from the equivariance of $\mathcal{T}_\tau$ (Proposition 7.1) or equivalently from the $\tau$-equivariance of any function $s_\tau + t\omega_C$ with $t > 0$. □

**Proposition 7.3.** The projection $\Pi_\tau$ defined by

$$ \Pi_{\tau}: \begin{cases} D_{\tau} \to \partial K(s_{\tau} + \omega_{C}) = S_{\tau}^{1} \\ X \mapsto P_{\tau}(X) + \mathcal{G}_{\Sigma_{C}}^{-1}(y_{\tau}(X)) \end{cases} $$

is continuous and $\Gamma_\tau$-equivariant, in the sense that for all $(\gamma, \tau_\gamma) \in \Gamma_\tau$,

$$ \Pi_{\tau}(\gamma X + \tau_{\gamma}) = \gamma \Pi_{\tau}(X) + \tau_{\gamma}. $$

*Proof.* The continuity and equivariance are just consequences of, respectively, Propositions 6.9 and 7.1. □

**Corollary 7.4.** There is a homeomorphism

$$ \Theta: \begin{cases} D_{\tau} \to S_{\tau}^{1} \times \mathbb{R}_{+}^{*} \\ X \mapsto (\Pi_{\tau}(X), \mathcal{T}_{\tau}(X)) \end{cases} $$

satisfying that for all $(\gamma, \tau_\gamma) \in \Gamma_\tau$

$$ (7.1) \qquad \Theta(\gamma X + \tau_{\gamma}) = (\gamma \Pi_{\tau}(X) + \tau_{\gamma}, \mathcal{T}_{\tau}(X)). $$
*Proof.* The continuity of $\Theta$ and condition (7.1) are consequences of Proposition 7.3. To see that $\Theta$ is a homeomorphism just notice that its inverse map is explicitly expressed as

$$ \Theta^{-1} = \begin{cases} S_{\tau}^{1} \times \mathbb{R}_{+}^{*} & \rightarrow & D_{\tau} \\ (X, t) & \mapsto & P_{\tau}(X) + t \mathcal{G}_{\Sigma_{c}}^{-1}(y_{\tau}(X)) \end{cases} $$

which is continuous by Proposition 6.9. □

**Proposition 7.5.** *The action of $\Gamma_{\tau}$ on $D_{\tau}$ is free and properly discontinuous and there is a homeomorphism*

$$ D_{\tau}/\Gamma_{\tau} \simeq S_{\tau}^{1}/\Gamma_{\tau} \times \mathbb{R}. $$

*Proof.* Using the homeomorphism $\Theta$ from Corollary 7.4 this is just a consequence of the free and properly discontinuous action of $\Gamma_{\tau}$ on the $C^1$ hypersurface $S_{\tau}^1 = \partial K(s_{\tau} + \omega_c)$ proved in Proposition 5.9. □

**Lemma 7.6.** *If $K$ is a spacelike $\tau$-convex domain ($\partial K = \partial_{sp}K$) in $(\mathbb{R}^{d+1}, C)$, then $\partial K$ is a Cauchy surface of $(D_{\tau}, C)$. Moreover it is $\Gamma_{\tau}$-equivariantly homeomorphic to $S_{\tau}^1$.*

*Proof.* The fact that $\partial K$ is a Cauchy surface of $D_{\tau}$ is Corollary 5.19. An explicit $\Gamma_{\tau}$-equivariant homeomorphism between $\partial K$ and $S_{\tau}^1$ is given by the restriction $\Pi_{\tau|\partial K}$ of the projection from Proposition 7.3. □

**Remark 7.7.** In Lemma 7.6, the fact that $\partial K$ is $\Gamma_{\tau}$-equivariantly homeomorphic to $S_{\tau}^1$ can also be seen as a direct consequence of Proposition 2.7 on the quotient affine spacetime $(D_{\tau}/\Gamma_{\tau}, C)$ (which will be the focus of subsection 7.2).

**Theorem 7.8 (Theorem 1.1 (1)).** *The action of $\Gamma_{\tau}$ on $D_{\tau}$ is free and properly discontinuous and there is a homeomorphism*

$$ D_{\tau}/\Gamma_{\tau} \simeq \Omega/\Gamma \times \mathbb{R}. $$

*Proof.* By Proposition 5.11, there exist a $C_+^2$ $\tau$-convex $K$ with $\partial K = \partial_{sp}K$. Then the Gauss map $\mathcal{G}_K$ on $K$ gives a diffeomorphism $\partial K/\Gamma_{\tau} \approx \Omega^*/\Gamma$. Propositions 5.1 and 7.5, and Lemma 7.6 then conclude the proof. □

## 7.2. Spacetime structure of the quotient.
In this subsection, we prove the GHCC part of point (2) (we shall treat maximality in subsection 8.3) from Theorem 1.1.

As $\Gamma$, the linear part of $\Gamma_{\tau}$, preserves the cone $C$, the parallel proper convex cone structure of $(D_{\tau}, C)$ descends to the equiaffine manifold $D_{\tau}/\Gamma_{\tau}$, endowing it with an affine spacetime structure.

Proposition 7.1 indicates that the cosmological time, normal projection and projecting normal descends to maps $\mathcal{T}_{\tau}: D_{\tau}/\Gamma_{\tau} \rightarrow \mathbb{R}_{+}^{*}$, $P_{\tau}: D_{\tau}/\Gamma_{\tau} \rightarrow \partial_{sp}D_{\tau}/\Gamma_{\tau}$, and $y_{\tau}: D_{\tau}/\Gamma_{\tau} \rightarrow \Omega^*/\Gamma$ (that we abusively denoted in the same way).

**Theorem 7.9 (Theorem 1.1, GHCC part of (2)).** *The affine spacetime $(D_{\tau}/\Gamma_{\tau}, C)$ is a globally hyperbolic spacetimes admitting a locally uniformly convex (Definition 2.11) and compact Cauchy surface.*

*Proof.* In the proof of Proposition 5.11, we have shown that there exists a $C_+^2$ $\tau$-convex domain $K$ with $\partial K = \partial_{sp}K$. The $C_+^2$ nature of $K$ implies that $\partial K/\Gamma_{\tau}$ is a spacelike locally uniformly convex hypersurface in $(D_{\tau}/\Gamma_{\tau}, C)$. It is also compact as the Gauss map $\mathcal{G}_K$ descends to a diffeomorphism from $\partial K/\Gamma_{\tau}$ to $\Omega^*/\Gamma$ which is compact by Proposition 5.1. Finally, Corollary 5.19 descends to the quotient and implies that $\partial K/\Gamma_{\tau}$ is a Cauchy surface of $(D_{\tau}/\Gamma_{\tau}, C)$. □

**Theorem 7.10.** *The affine spacetime $(D_{\tau}/\Gamma_{\tau}, C)$ is foliated by the level hypersurfaces of its cosmological time, which are $C$-spacelike and $C^1$ embedded convex Cauchy surfaces.*
*Proof.* The fact that level hypersurfaces of the cosmological time are a $C^1$ foliation is just a quotiented version of Proposition 7.2. Those are quotients of $C$-spacelike convex hypersurfaces, and thus, by Corollary 5.19, are convex Cauchy surfaces of $(D_\tau/\Gamma_\tau, C)$. □

*Remark 7.11.* Remembering the discussion in subsection 5.6, we can consider the two GHCC affine spacetimes $(D_\tau^C/\Gamma_\tau, C)$ and $(D_\tau^{-C}/\Gamma_\tau, -C)$. In general, they have no reason to be isomorphic affine spacetimes. In the case where $d+1$ is even and $\tau = 0$, they actually are: indeed $-\mathrm{Id} \in \mathrm{SL}(\mathbb{R}^{d+1})$ and we have (see subsection 5.6) that $D_0^C = -D_0^{-C}$.

In the case of flat Lorentzian spacetimes [Mes07, Bon05], $-\mathrm{Id}$ never belongs to $\mathrm{SO}_0(d, 1) \subset \mathrm{SL}(\mathbb{R}^{d+1})$, moreover one can arbitrarily fix a time orientation on the common ambient Minkowski space containing all the maximal domains (the cone $C$ does not vary) in order to have a favourite choice between the two spacetimes given by the two quotients of those maximal domains.

## 8. CORRESPONDENCE BETWEEN AFFINE DEFORMATIONS OF DIVIDING GROUPS AND AFFINE SPACETIMES

This section is devoted to the proofs of the remaining maximality part of Theorem 1.1 (2) and of Theorem 1.2, through the study of the developing maps of *Globally Hyperbolic affine spacetimes admitting a locally uniformly Convex and Compact Cauchy surface* (GHCC affine spacetimes).

### 8.1. Restriction of a developing map to a compact locally uniformly convex Cauchy surface.
In this subsection, we study the restriction of developing maps of GHCC affine spacetimes to a compact locally uniformly convex Cauchy surface.

Let $(M, C^M)$ be a GHCC affine spacetime. Using the equiaffine structure of $M$, we choose a pair $(\mathrm{dev}, \rho)$ of developing and holonomy maps (see subsection 2.1). The developing map $\mathrm{dev}: \tilde{M} \to A^{d+1}$, is then a local diffeomorphism and it is $\rho$-equivariant, that is for all $p \in \tilde{M}$ and $\gamma \in \pi_1 M$,

$$ \mathrm{dev}(\gamma \cdot p) = \rho(\gamma) \cdot \mathrm{dev}(p). $$

The cone structure $C^M$ on $M$ lifts to a cone structure $C_{\tilde{M}}$ on $\tilde{M}$ which is parallel for the flat connection induced by the equiaffine structure of $\tilde{M}$. Thus, we can set $C \subset \mathbb{V}^{d+1}$ to be the proper open convex cone such that in each tangent space $T_{\mathrm{dev}(p)}A^{d+1} = \mathbb{V}^{d+1}$, we have

$$ C = d_p \mathrm{dev}(C_{\tilde{M}}), $$

from now on we shall abusively write that this cone is defined as $C := d \mathrm{dev}(C^M)$.

Let $S$ be a compact locally uniformly convex Cauchy surface of $M$. Remember that Proposition 2.7 implies that $\pi_1 M = \pi_1 S$ and that $\tilde{S}$, the preimage of $S$ in the universal covering $\tilde{M} \to M$, is a universal cover of $S$.

By definition of a locally uniformly convex Cauchy surface (Definition 2.11), the developing map $\mathrm{dev}$ induces a $C$-spacelike locally uniformly convex immersion $\mathrm{dev}_{|\tilde{S}}: \tilde{S} \to A^{d+1}$, namely $\mathrm{dev}_{|\tilde{S}}$ is locally a diffeomorphism onto a $C$-spacelike graph of a uniformly convex $C^2$ map. We can thus consider the Blaschke metric $h$ on $\tilde{S}$ induced by that immersion (Definition 3.4). We can also fix any Euclidean metric $g_{euc}$ on $A^{d+1}$ and consider its pullback by the restriction $\mathrm{dev}_{|\tilde{S}}$, denoted by $\mathrm{dev}_{|\tilde{S}}^*g_{euc}$.

As the immersion $\mathrm{dev}_{|\tilde{S}}: \tilde{S} \to A^{d+1}$ is $\rho$-equivariant and $\rho$ takes values in $\mathrm{SA}(A^{d+1})$, the Blaschke metric $h$ on $\tilde{S}$ is preserved by $\pi_1 S$ and descends to a Riemannian metric $\tilde{h}$ on $S = \tilde{S}/\pi_1 S$. The Riemannian manifold $(S, \tilde{h})$ is closed and thus complete, so $(\tilde{S}, h)$ is also complete. We are thus in the setting of the following theorem of Trudinger and Wang [TW02, Theorem A].
**Theorem 8.1** (Trudinger-Wang [TW02]). Let $i: M \to A^{d+1}$ be a locally uniformly convex immersed hypersurface, namely the map $i$ is locally a diffeomorphism onto a graph (in some well chosen affine coordinates) of a uniformly convex $C^2$ function. Let $h$ be the Blaschke metric on $M$ induced by that immersion and $g_{euc}$ be a Euclidean metric on $A^{d+1}$. If the Riemannian manifold $(M, h)$ is complete, then $(M, i^*g_{euc})$ is also complete.

We can also parametrise $A^{d+1}$ as $\mathbb{R}^{d+1}$ so that we have

$$C = \{t(x, 1) \mid x \in \Omega, t > 0\},$$

where $\Omega$ is a bounded open convex domain of $\mathbb{R}^d$ containing 0. In that setting, the following lemma holds.

**Lemma 8.2.** Let $C \subset \mathbb{V}^{d+1}$ be an open proper convex cone and $g_{euc}$ a Euclidean metric on $A^{d+1}$. Parametrise $A^{d+1}$ as $\mathbb{R}^{d+1}$ such that $C = \{t(x, 1) \mid x \in \Omega, t > 0\}$ where $\Omega$ is a bounded open convex domain of $\mathbb{R}^d$ containing 0, and let $\pi: \mathbb{R}^{d+1} \to \mathbb{R}^d$ be the projection onto the first $d$ coordinates. If $i: M \to A^{d+1}$ is a $C$-spacelike immersed hypersurface, then the map $\pi \circ i: (M, i^*g_{euc}) \to (\mathbb{R}^d, g_{euc}|_{\mathbb{R}^d})$ is bi-Lipschitz.

*Proof.* As $\Omega$ is a bounded open convex domain containing 0, we can set

$$M := \sup \{g_{euc}|_{\mathbb{R}^d}(x, x) \mid x \in \Omega\} > 0.$$

For every $C$-spacelike vector $V \in \mathbb{R}^{d+1}$, we then have

$$g_{euc}|_{\mathbb{R}^d}(\pi(V), \pi(V)) \le g_{euc}(V, V) \le \left(1 + \frac{1}{M}\right) g_{euc}|_{\mathbb{R}^d}(\pi(V), \pi(V)).$$

Applying that inequality to vectors $V$ tangent to the immersed hypersurface $i: M \to A^{d+1}$ we get the lemma. □

We can now prove the following.

**Proposition 8.3.** Let $(\mathrm{dev}, \rho)$ be a pair of developing and holonomy maps of a GHCC affine spacetime $(M, C^M)$ and parametrise $A^{d+1}$ as $\mathbb{R}^{d+1}$ so that the open proper convex cone $C := \mathrm{d} \mathrm{dev}(C^M)$ is expressed as $C = \{t(x, 1) \mid x \in \Omega, t > 0\}$, where $\Omega$ is a bounded open convex domain of $\mathbb{R}^d$ containing 0. If $S$ is a compact locally uniformly convex Cauchy surface of $(M, C^M)$, the developing map $\mathrm{dev}$ on $\tilde{M}$ embeds its universal cover $\tilde{S} \subset \tilde{M}$ into $\mathbb{R}^{d+1}$ as the graph of a $C$-spacelike and locally uniformly convex function $f: \mathbb{R}^d \to \mathbb{R}$.

*Proof.* Using the definition of a locally uniformly convex Cauchy surface (Definition 2.11) and our well chosen parametrisation of $A^{d+1}$, we already know that the image of $\mathrm{dev}|_{\tilde{S}}$ is locally the graph of a $C$-spacelike and locally uniformly convex function of the first $d$ coordinates. By Lemma 8.2, $\pi \circ \mathrm{dev}|_{\tilde{S}}$ is injective, thus all those local graphs agree with each other. That means that $\mathrm{dev}(\tilde{S})$ is the graph of a $C$-spacelike locally uniformly convex function $f: U \to \mathbb{R}$ on some open subset $U \subseteq \mathbb{R}^d$.

Now, let us show that $U$ is the whole $\mathbb{R}^d$. Remember that the Blaschke metric $h$ on $\tilde{S}$ given by the immersion $\mathrm{dev}|_{\tilde{S}}: \tilde{S} \to \mathbb{R}^{d+1}$ descends to the closed manifold $S = \tilde{S}/\pi_1 S$, so that $(\tilde{S}, h)$ is complete. By Theorem 8.1, $(\tilde{S}, \mathrm{dev}^*_{\tilde{S}}g_{euc})$ is also complete. As bi-Lipschitz maps preserve completeness, Lemma 8.2 implies that $(U = \pi \circ \mathrm{dev}(\tilde{S}), g_{euc}|_{\mathbb{R}^d})$ is complete, so that $U$ is the whole $\mathbb{R}^d$.

Finally, let us prove that $\mathrm{dev}|_{\tilde{S}}: \tilde{S} \to \mathbb{R}^{d+1}$ is an embedding. By Lemma 8.2, $\pi \circ \mathrm{dev}|_{\tilde{S}}$ is injective on $\tilde{S}$, thus $\mathrm{dev}$ is also injective on $\tilde{S}$. As $(\tilde{S}, \mathrm{dev}^*_{\tilde{S}}g_{euc})$ is complete, the Hopf-Rinow Theorem implies that $\mathrm{dev}|_{\tilde{S}}$ is proper. Thus $\mathrm{dev}|_{\tilde{S}}$ is a proper injective immersion, that is $\mathrm{dev}|_{\tilde{S}}$ is an embedding. □

As we parametrised $A^{d+1}$ as $\mathbb{R}^{d+1}$, we can also see $\mathrm{SA}(A^{d+1})$ as $\mathrm{SA}(\mathbb{R}^{d+1}) = \mathrm{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$. Consider the linear part of the representation $\rho: \pi_1 S \to \mathrm{SA}(\mathbb{R}^{d+1})$, it is a representation $\rho_{\mathrm{lin}}: \pi_1 S \to \mathrm{SL}(\mathbb{R}^{d+1})$. Then the following holds.
**Proposition 8.4.** In setting of Proposition 8.3, let $\mathcal{G}: \mathrm{dev}(\tilde{S}) \to \Omega^*$ be the Gauss map of the $\mathcal{C}$-spacelike hypersurface graph($f$) = $\mathrm{dev}(\tilde{S})$. The map $\phi := \mathcal{G} \circ \mathrm{dev}: \tilde{S} \to \Omega^*$ is a diffeomorphism onto its image $\Theta^* := \mathcal{G} \circ \mathrm{dev}(\tilde{S})$, which is an open convex subset of $\Omega^*$ preserved by the dual projective action (described at the beginning of Section 5) of $\rho_{\mathrm{lin}}(\pi_1 S)$, the linear part of $\rho$ on $\Omega^* \simeq \mathbb{P}(\mathcal{C}^*)$. Moreover, it is $\rho_{\mathrm{lin}}$-equivariant, that is for all $p \in \tilde{S}$ and $\gamma \in \pi_1 S$,

$$ \phi(\gamma \cdot p) = \rho_{\mathrm{lin}}(\gamma) * \phi(p). $$

*Proof.* As graph($f$) is a $\mathcal{C}$-spacelike locally uniformly convex hypersurface, the Gauss map $\mathcal{G}$ is defined as

$$ \mathcal{G}: \left\{ \begin{array}{rcl} \mathrm{graph}(f) & \to & \Omega^* \\ (x, f(x)) & \mapsto & \mathrm{grad} \, f(x) \end{array} \right. $$

and is a $\mathcal{C}^1$ diffeomorphism onto its image $\Theta^*$ which is an open subset of $\Omega^*$. This subset $\Theta$ is the interior of the support of the convex function $f^*$, the Legendre-Fenchel transform of $f$, and thus it is convex.

By Proposition 8.3, the restriction $\mathrm{dev}|_{\tilde{S}}$ is a diffeomorphism onto graph($f$) so that $\phi = \mathcal{G} \circ \mathrm{dev}$ is a diffeomorphism onto $\Theta^*$.

As $\tilde{S}$ is preserved by $\pi_1 S = \pi_1 M$ and $\mathrm{dev}$ is $\rho$-equivariant, $\mathrm{dev}(\tilde{S}) = \mathrm{graph}(f)$ is preserved by $\rho(\pi_1 S)$. Using the equivariance of the Gauss map $\mathcal{G}$, we get that $\Theta^* = \mathcal{G} \circ \mathrm{dev}(\tilde{S})$ is preserved by $\rho_{\mathrm{lin}}(\pi_1 S)$ and that $\phi = \mathcal{G} \circ \mathrm{dev}$ is $\rho_{\mathrm{lin}}$-equivariant. $\square$

**Corollary 8.5.** Let $\rho: \pi_1 M \to \mathrm{SL}(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ be a holonomy map of a GHCC affine spacetime $(M, \mathcal{C}^M)$. The linear part of $\rho$ is an injective representation $\rho_{\mathrm{lin}}: \pi_1 M \to \mathrm{SL}(\mathbb{R}^{d+1})$. That means, by Remark 5.4, that $\rho(\pi_1 M)$ is an affine deformation of its linear part $\rho_{\mathrm{lin}}(\pi_1 S)$.

*Proof.* As $S$ is a closed manifold, the action of $\pi_1 S$ on $\tilde{S}$ is free, and thus using the $\rho_{\mathrm{lin}}$-equivariance of the diffeomorphism $\phi$ between $\tilde{S}$ and $\Theta^*$ (Proposition 8.4), $\rho_{\mathrm{lin}}$ must be injective. $\square$

**Corollary 8.6.** In the setting of Proposition 8.4, the subgroup $\rho_{\mathrm{lin}}(\pi_1 S) < \mathrm{SL}(\mathbb{R}^{d+1})$ is discrete and the open convex domain $\Theta^*$, seen as a projective convex domain inside $\Omega^* \simeq \mathbb{P}(\mathcal{C}^*)$, is divisible by the dual projective action of $\rho_{\mathrm{lin}}(\pi_1 S)$.

*Proof.* As $S$ is a closed manifold, the action of $\pi_1 S$ on $\tilde{S}$ is free and properly discontinuous, and thus using the $\rho_{\mathrm{lin}}$-equivariance of the diffeomorphism $\phi$ between $\tilde{S}$ and $\Theta^*$ (Proposition 8.4), we get that the dual projective action of $\rho_{\mathrm{lin}}(\pi_1 S)$ on $\Theta^*$ is free and properly discontinuous so $\rho_{\mathrm{lin}}(\pi_1 S)$ is discrete. Moreover we also have that the quotients $S = \tilde{S}/\pi_1 S$ and $\Theta^*/\rho_{\mathrm{lin}}(\pi_1 S)$ are diffeomorphic. As $S$ is compact, that means that $\Theta^*$ is divisible by $\rho_{\mathrm{lin}}(\pi_1 S)$. $\square$

Corollary 8.6 indicates that we are in a setting where we can apply Lemma 5.2 to $\Theta^* \subseteq \Omega^*$. That implies that $\Theta^* = \Omega^*$, i.e. that the Gauss map $\mathcal{G}$ in Proposition 8.4 is surjective. Thus the epigraph of $f$ is a $\rho(\pi_1 M)$-invariant $\mathcal{C}$-convex domain (Definition 4.2). As $\rho_{\mathrm{lin}}(\pi_1 M) = \rho_{\mathrm{lin}}(\pi_1 S)$ divides $\mathcal{C}^*$, the cone over $\Omega^*$, by Proposition 5.1, it also divides $\mathcal{C}$. To summarise, we have the following.

**Proposition 8.7.** Let $(\mathrm{dev}, \rho)$ be pair of developing and holonomy maps of a GHCC affine spacetime $(M, \mathcal{C}^M)$. Parametrise $\mathbb{A}^{d+1}$ as $\mathbb{R}^{d+1}$ and consider the proper convex cone $\mathcal{C} := \mathrm{d} \mathrm{dev}(\mathcal{C}^M) \subset \mathbb{R}^{d+1}$. The holonomy map $\rho$ is injective and its image is an affine deformation of its linear part $\Gamma := \rho_{\mathrm{lin}}(\pi_1 M)$, which divides the cone $\mathcal{C}$. Introducing $\tau: \Gamma \to \mathbb{R}^{d+1}$ the cocycle such that $\rho(\pi_1 M) = \Gamma_\tau$, if $S$ is a compact locally uniformly convex Cauchy surface of $(M, \mathcal{C}^M)$, the developing map $\mathrm{dev}$ embeds $\tilde{S}$ in $\mathbb{R}^{d+1}$ as the boundary of a $\mathcal{C}$-spacelike and $\mathcal{C}_+^2$-convex domain.

8.2. Properties of developing maps on GHCC affine spacetimes. In this subsection, we extend the results in Proposition 8.7 to properties on developing maps of GHCC affine spacetimes, in order to prove Theorem 8.10, which is a classification theorem for GHCC implying Theorem 1.2.
**Proposition 8.8.** Let $(\mathrm{dev}, \rho)$ be a pair of developing and holonomy maps of a GHCC affine spacetime $(M, C^M)$, parametrise $A^{d+1}$ as $\mathbb{R}^{d+1}$, consider the proper convex cone $C := \mathrm{d} \mathrm{dev}(C^M) \subset \mathbb{R}^{d+1}$ and, by Proposition 8.7, introduce $\tau$ a cocycle such that $\rho(\pi_1 M) = \Gamma_\tau$ is an affine deformation of the subgroup $\Gamma := \rho_{\mathrm{lin}}(\pi_1 M)$ dividing $C$. The image $\mathrm{dev}(\widetilde{M})$ is contained in the maximal $\tau$-convex domain $D_\tau$ of $(\mathbb{R}^{d+1}, C)$.

*Proof.* Let us set $S$ to be compact locally uniformly convex Cauchy surface of $M$.

First notice that, using Proposition 5.18 and Proposition 8.7, we have $D(\mathrm{dev}(\tilde{S})) = D_\tau$.

Now, let $X = \mathrm{dev}(p) \in \mathrm{dev}(\widetilde{M})$ and $X(t)$ be an inextensible $C$-causal curve in $\mathbb{R}^{d+1}$ parametrised by $t \in (-\infty, +\infty)$ and such that $X(0) = X$. As $\mathrm{dev}$ is a local diffeomorphism around $p$, we have that locally around $t=0$ the path $X(t)$ lifts to a causal curve $p(t)$ in $\widetilde{M}$ such that $\mathrm{dev}(p(t)) = X(t)$.

Let $t^- \in \mathbb{R}_-^* \cup \{-\infty\}$ and $t^+ \in \mathbb{R}_+^* \cup \{+\infty\}$ be such that $(t^-, t^+)$ is the maximal interval on which the developing map lifts $X(t)$ to a causal curve $p(t)$. By definition of $t^-$ and $t^+$, $\{p(t) | t \in (t^-, t^+)\}$ is an inextensible causal curve in $\widetilde{M}$, thus there exist a unique $t_s \in (t^-, t^+)$ such that $p(t_s)$ belongs to the Cauchy surface $\tilde{S}$. Then, $\mathrm{dev}(p(t_s)) = X(t_s) \in \mathrm{dev}(\tilde{S})$.

Thus, any inextensible $C$-causal curve going through $X = \mathrm{dev}(p)$ intersects $\mathrm{dev}(\tilde{S})$, meaning precisely that $\mathrm{dev}(p) \in D(\mathrm{dev}(\tilde{S})) = D_\tau$. $\square$

**Proposition 8.9.** A developing map $\mathrm{dev} : \widetilde{M} \to A^{d+1}$ on a GHCC affine spacetime $(M, C^M)$ is necessarily injective.

*Proof.* Let us work in the setting of Proposition 8.8 and prove it by contradiction. Assume there are $p \neq q \in \widetilde{M}$ such that $\mathrm{dev}(p) = \mathrm{dev}(q) = X \in D_\tau$ (by Proposition 8.8). Let $X(t)$ be an inextensible $C$-causal curve with $X(0) = X$. As $X \in D_\tau = D(\mathrm{dev}(\tilde{S}))$, there is a unique $t_s \in \mathbb{R}$ such that $X(t_s) \in \mathrm{dev}(\tilde{S})$. Following what was done in the proof of Proposition 8.8, there are real numbers $t^- < \min(0, t_s)$ and $t^+ > \max(0, t_s)$ such that for all $t \in (t^-, t^+)$,

$$X(t) = \mathrm{dev}(p(t)) = \mathrm{dev}(q(t)),$$

where $p(t), q(t)$ are causal curves in $\widetilde{M}$ and $p(t_s), q(t_s) \in \tilde{S}$. As

$$X(t_s) = \mathrm{dev}(p(t_s)) = \mathrm{dev}(q(t_s)),$$

using the injectivity of the developing map on $\tilde{S}$ (Proposition 8.7), one gets that $p(t_s) = q(t_s)$ (thus $t_s \neq 0$).

If $t_s < 0$, let $T = \max\{t \in [t_s, 0] | p(t) = q(t)\}$ which is well-defined because of the continuity of $p(t)$ and $q(t)$ and that $p(t_s) = q(t_s)$. By assumption $p(0) \neq q(0)$ and thus $t_s \le T < 0$. Then for all $\varepsilon > 0$ such that $T < T + \varepsilon < 0$, we have

$$p(T + \varepsilon) \neq q(T + \varepsilon) \quad \text{and} \quad X(T + \varepsilon) = \mathrm{dev}(p(T + \varepsilon)) = \mathrm{dev}(q(T + \varepsilon)),$$

which contradicts the fact that $\mathrm{dev}$ is a local diffeomorphism at $p(T) = q(T)$ (see Figure 8.1).

The case where $t_s > 0$ can be treated in a similar way. $\square$

**Theorem 8.10.** Let $(M, C^M)$ be a GHCC affine spacetime. Any pair $(\mathrm{dev}, \rho)$ of a developing map

$$\mathrm{dev} : \widetilde{M} \to \mathbb{R}^{d+1},$$

and a holonomy map

$$\rho : \pi_1 M \to SA(\mathbb{R}^{d+1}),$$

given by its equiaffine structure satisfies the following:

* the holonomy map $\rho$ is a group isomorphism onto its image $\rho(\pi_1 M)$ which is an affine deformation $\Gamma_\tau$ of a subgroup $\Gamma < SL(\mathbb{R}^{d+1})$ dividing the cone $C := \mathrm{d} \mathrm{dev}(C^M)$,
<figure><img src="image_13.png" /><figcaption>**Figure 8.1.** Illustration of the situation in the proof of Proposition 8.9.</figcaption></figure>

• the developing map dev is an isomorphism of affine spacetimes onto its image which is included in $(D_{\tau}, \mathcal{C})$, where $D_{\tau}$, the domain associated with $\Gamma_{\tau} := \rho(\pi_{1}M)$ and $\mathcal{C}$, is given by Theorem 1.1,

• the developing map dev descends to an embedding $(M, C^M) \hookrightarrow (D_\tau/\Gamma_\tau, C)$ preserving the equiaffine and cone structures, and mapping any locally uniformly convex and compact Cauchy surface of $(M, C^M)$ to a locally uniformly convex and compact Cauchy surface of $(D_\tau/\Gamma_\tau, C)$.

*Proof.* The first point is a consequence of Proposition 8.7 and the second one of Propositions 8.8 and 8.9. Finally, using those two first points and the equivariance of the developing map we get the third one. □

In Theorem 8.10, if moreover $(M, C^M)$ is a MGHCC, then the definition of maximality of globally hyperbolic affine spacetimes (Definition 2.10) implies that the embedding $(M, C^M) \hookrightarrow (D_\tau/\Gamma_\tau, C)$ given by the developing map must be surjective, proving Theorem 1.2.

**8.3. Maximality of the quotient spacetimes $D_\tau/\Gamma_\tau$.** In this subsection, we use the preceding work in order to prove the maximality of spacetimes built with Theorem 1.1.

Let $\Gamma_\tau < SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ be an affine deformation of a group $\Gamma < SL(\mathbb{R}^{d+1})$ dividing an open proper convex cone $\mathcal{C}$. We consider the associated spacetimes $(D_\tau^C/\Gamma_\tau, \mathcal{C})$ given by Theorem 1.1. In Section 7 we have proved that it is a GHCC affine spacetime. Using the results from subsection 8.2 we have the following.

**Theorem 8.11** (Theorem 1.1 (2)). Let $\Gamma_\tau < SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ is an affine deformation of a group $\Gamma < SL(\mathbb{R}^{d+1})$ dividing an open proper convex cone $\mathcal{C}$. The GHCC affine spacetime $(D_\tau/\Gamma_\tau, \mathcal{C})$ is a maximal globally hyperbolic spacetime.

*Proof.* Let $i : (D_\tau/\Gamma_\tau, \mathcal{C}) \hookrightarrow (M', C')$ be an embedding preserving the equiaffine and cone structures, and mapping a locally uniformly convex compact Cauchy surface $S$ of $D_\tau/\Gamma_\tau$ to a Cauchy surface $S'$ of $M'$.

First, notice that, because $i$ preserves the equiaffine structure, $S'$ is also a locally uniformly convex Cauchy surface and thus $(M', C')$ is also a GHCC affine spacetime. It lifts to an embedding $\tilde{i} : (D_\tau, \mathcal{C}) \hookrightarrow (\tilde{M}', C')$ also preserving the equiaffine and cone structures, and mapping a locally uniformly convex Cauchy surface $\tilde{S}$ of $D_\tau$ to a locally uniformly convex Cauchy surface $\tilde{S}'$ of $\tilde{M}'$. We want to show that $i$ is surjective.

Now, let $(\mathrm{dev}, \rho)$ be a couple of developing and holonomy maps for the the GHCC affine spacetime $(M', C')$. Theorem 8.10 states that, $\rho(\pi_1 M')$ is an affine deformation $\Gamma_{\tau''}''$ of its linear part $\Gamma'' < SL(\mathbb{R}^{d+1})$, which divides a proper convex cone $C''$. It also states that the developing map dev on $(\tilde{M}', C')$ embeds it into $(D_{\tau''}''/\Gamma_{\tau''}''', C'')$.

Consider the following embedding $\tilde{j} = \mathrm{dev} \circ \tilde{i} : (D_\tau, \mathcal{C}) \hookrightarrow (D_{\tau''}, C'')$, which descends to an embedding $j : (D_\tau/\Gamma_\tau, \mathcal{C}) \hookrightarrow (D_{\tau''}/\Gamma_{\tau''}, C'')$. It preserves the equiaffine structure between two convex domains of
$\mathbb{R}^{d+1}$ so it is of the form $\tilde{j} : X \to AX$, where $A \in \text{SL}(\mathbb{R}^{d+1})$. Thus, up to post-composing we can assume that $\tilde{j}$ is a restriction of the identity.

As the embedding $\tilde{j}$ is a restriction of the identity, we clearly have $D_{\tau} \subseteq D_{\tau''}$. Moreover as $\tilde{j}$ preserves the cone structure, we also have $C = C''$.

Using the immersion $i : (D_{\tau}/\Gamma_{\tau}, C) \to (M', C')$ one can see $\pi_1(D_{\tau}/\Gamma_{\tau}) \simeq \Gamma_{\tau}$ as included in $\pi_1 M'$. Actually, as $S$ is a Cauchy surface of $M$ and $S' = i(S)$ is a Cauchy surface of $M'$, by Proposition 2.7, we have an identification

$$ \Gamma_{\tau} \simeq \pi_1(D_{\tau}/\Gamma_{\tau}) \simeq \pi_1 S \simeq \pi_1 S' \simeq \pi_1 M'. $$

Then, using that identification, the embedding $\tilde{j} = \text{dev} \circ \tilde{i}$ is $\rho$-equivariant in the sense that for all $(\gamma, \tau_{\gamma}) \in \Gamma_{\tau} \simeq \pi_1 M'$ and $X \in D_{\tau}$, we have

$$ \tilde{j}(\gamma X + \tau_{\gamma}) = \rho(\gamma, \tau_{\gamma}) \cdot \tilde{X}, $$

where $\rho(\gamma, \tau_{\gamma}) \in \text{SA}(\mathbb{R}^d)$. As $\tilde{j}$ is a restriction of the identity and $D_{\tau}$ is open, we get that the affine transformations $(\gamma, \tau_{\gamma})$ and $\rho(\gamma, \tau_{\gamma})$ are equal. Thus $\Gamma''_{\tau''} = \rho(\pi_1 M') = \Gamma_{\tau}$.

Finally, that implies that $\tilde{j} = \text{Id}|_{D_{\tau}}$ is surjective. Hence, as $\tilde{j} = \text{dev} \circ \tilde{i}$ and $\text{dev}$ is a diffeomorphism onto its image, $\tilde{i}$ is also surjective and so is $i$. $\square$

**Remark 8.12.** Let $(M, C)$ be a MGHCC affine spacetime, like in subsection 6.2, using affine spheres in the tangent spaces of $M$, one can again define a norm $F$ on $C$-null and $C$-timelike vectors, a length $l_F$ of $C$-causal curve and a $C$-distance $\rho$ satisfying the time inequality. Theorem 1.2 implies that again this norm endows $(M, C)$ with a timelike space structure in the sense of Busemann [Bus67], and that the cosmological time on $(M, C)$ satisfies

$$ \begin{aligned} \mathcal{T}(p) &= \sup \{ \rho(q, p) \mid q \in J^{-}(p) \}, \\ &= \sup \{ l_{F}(\gamma) \mid \gamma \text{ is a future directed } C\text{-causal curve from a point of } J^{-}(p) \text{ to } p \}. \end{aligned} $$

# APPENDIX A. EXAMPLES OF DIVIDING GROUPS ADMITTING AFFINE DEFORMATIONS

This appendix is devoted to the construction of examples proving the following proposition.

**Proposition A.1.** In any dimension $d \ge 2$, there exists a discrete subgroup $\Gamma < \text{SL}(\mathbb{R}^{d+1})$ dividing a proper convex cone $C \subset \mathbb{R}^{d+1}$ over a proper convex domain $\Omega \subset \mathbb{P}(\mathbb{R}^{d+1})$ which is not projectively equivalent to an ellipse (giving the Klein ball model of the hyperbolic space).

**Bending of holonomy groups.** An amalgamated product $\Gamma = \Gamma' *_{\Lambda} \Gamma'' < \text{SL}(\mathbb{R}^{d+1})$ can be algebraically deformed through the following general procedure called *bending* and deeply studied by Johnson and Millson [JM87]. Let $(A_s)_{s \in \mathbb{R}}$ be a continuous path of elements of $\text{SL}(\mathbb{R}^{d+1})$ such that $A_0$ is the identity and every $A_s$ commutes with all the elements of $\Lambda$. Then the bending procedure consists in continuously deforming $\Gamma$ by looking at the family $\Gamma_s = \Gamma' *_{\Lambda} (A_s \Gamma'' A_s^{-1})$.

**Bulging the holonomy groups of hyperbolic manifolds.** Let $\Gamma_0 < \text{SO}_0(d, 1)$ be the holonomy group of a closed hyperbolic $d$-dimensional manifold $M = \mathbb{B}^d/\Gamma_0$ (using the Klein ball projective model of the hyperbolic space) containing a totally geodesic embedded compact connected hypersurface $N$ separating $M$ into two connected components: $M \setminus N = M' \cup M''$. We shall deform $\Gamma_0$ through a particular bending procedure called *bulging* (following the nomenclature in [BV24]).

Using the inclusion $i : \pi_1 N \to \pi_1 M$ and taking the image by the holonomy of $\pi_1 N$ we get a subgroup $\Lambda < \Gamma_0$ preserving a hyperplane $H$ intersecting the Minkowski light cone $C_0$. Then $H \cap (\mathbb{B}^d \times \{1\})$ is a connected preimage of $N$ by the projection $\mathbb{B}^d \times \{1\} \to M = \mathbb{B}^d/\Gamma_0$ and a universal cover over $N$. For $k \in \{1, 2\}$, using the inclusion $i^{(k)} : \pi_1 M^{(k)} \to \pi_1 M$ and setting $\Gamma_0^{(k)}$ to be the holonomy of $\pi_1 M^{(k)}$,
by the Seifert–van Kampen Theorem we can see the holonomy group of M as the amalgamated product $\Gamma_0 = \Gamma'_0 *_{\Lambda} \Gamma''_0$.

Let $X \in \mathbb{R}^{d,1} = \mathbb{R}^{d+1}$ be a unit Minkowski spacelike vector orthogonal to the hyperplane $H$. For later constructions, it is worth noting that $X$ is a common eigenvector, with associated eigenvalue 1, of all elements of $\Lambda$. For $s \in \mathbb{R}$, let $A_s \in SL(\mathbb{R}^{d+1})$ be the element that restricts to the homothety $e^s I_n$ on $H$ and sends $X$ to $e^{-ds}X$ (in a well-chosen basis it has form $\text{Diag}(e^s, \dots, e^s, e^{-ds})$ and thus its determinant is indeed 1). As $A_s \gamma A_s^{-1} = \gamma$, for all $\gamma \in \Lambda$, we can continuously deform $\Gamma_0$ by bulging by considering $\Gamma_s := \Gamma'_0 *_{\Lambda} (A_s \Gamma''_0 A_s^{-1}) < SL(\mathbb{R}^{d+1})$. Then, by the work of Koszul [Kos68] and Benoist [Ben05], each group $\Gamma_s$ happens to also divide an open proper convex cone $C_s$, which is not affinely equivalent to the Minkowski light cone if $s \neq 0$ [Ben08, Section 8.2].

**Bending to get affine deformations of groups obtained by bulging.** Let us now prove that such a group $\Gamma := \Gamma_{s_0}$ (we fix $s_0$ for the rest of this section) obtained by bulging admits a non-trivial cocycle $\tau \in H^1(\Gamma, \mathbb{R}^{d+1})$, i.e. that $\Gamma$ admits a non-trivial affine deformation $\Gamma_{\tau} < SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$. We shall proceed by seeing $SL(\mathbb{R}^{d+1})$ as included in $SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ in order to bend the amalgamated product $\Gamma = \Gamma' *_{\Lambda} \Gamma''$ (where $\Gamma' = \Gamma'_0$ and $\Gamma'' = A_s \Gamma''_0 A_s^{-1}$).

Remembering that there is $X \in \mathbb{R}^{d+1}$, a common eigenvector, with associated eigenvalue 1, of all elements of $\Lambda$, we set for all $s \in \mathbb{R}$, $B_s = (I, sX) \in SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$, where $I$ is the identity in $SL(\mathbb{R}^{d+1})$. Then notice that for all $\gamma \in \Gamma$ and $s \in \mathbb{R}$,

$$ (A.1) \qquad B_s(\gamma, 0)B_s^{-1} = (\gamma, s(X - \gamma X)). $$

If moreover $\gamma \in \Lambda$, as $X$ is an eigenvector of $\gamma$ with associated eigenvalue 1, it becomes

$$ B_s(\gamma, 0)B_s^{-1} = (\gamma, 0). $$

Thus, we can consider the bending of $\Gamma = \Gamma_{\tau=0} < SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ by the family $(B_s)_{s \in \mathbb{R}}$. We get a family of groups $G_s = \Gamma' *_{\Lambda} (B_s \Gamma'' B_s^{-1})$ in $SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$. Equality (A.1) implies that we have actually built a family of affine deformations $(\Gamma_{\tau_s})_{s \in \mathbb{R}}$ of $\Gamma$.

To conclude, let us prove that for $s \neq 0$, such a deformation is non-trivial. It suffices to prove it for $s = 1$. In order to lighten expressions we shall denote $\tau_1$ by $\tau$. By contradiction, assume that $\tau$ is a coboundary, that is that we can set $V \in \mathbb{R}^{d+1}$ such that for all $\gamma \in \Gamma$, $\tau(\gamma) = (I - \gamma)V$. For all $\gamma' \in \Gamma'$, we have

$$ \tau(\gamma') = (I - \gamma')V = 0, $$

thus $V$ is either 0 or an eigenvector of all $\gamma' \in \Gamma'$ with associated eigenvalue 1.

As $\Lambda$ is conjugate to a subgroup of the form

$$ \left( \begin{array}{c|c} G & 0 \\ \hline 0 & 1 \end{array} \right) < SL(\mathbb{R}^{d+1}) $$

where $G < SO(d-1,1)$ is the holonomy group of a closed hyperbolic $(d-1)$-dimensional manifold $N$, its only common eigenvector is $X$. Thus, $V$ is necessarily in the span of $X$.

We claim that for all $\gamma \in \Gamma \setminus \Lambda$, $X$ is not an eigenvector of $\gamma$ with associated eigenvalue 1. Then, for $\gamma' \in \Gamma' \setminus \Lambda$, as $X$ is not an eigenvector with associated eigenvalue 1 of $\gamma'$, if $V \neq 0$, then

$$ \tau(\gamma') = (I - \gamma')V \neq 0, $$

so necessarily $V = 0$. But then for $\gamma'' \in \Gamma'' \setminus \Lambda$, as $X$ is not an eigenvector with associated eigenvalue 1 of $\gamma''$,

$$ \tau(\gamma'') = (I - \gamma'')X \neq 0, $$
and thus $V \neq 0$. That is a contradiction.

Now let us prove our claim. Let $\gamma \in \Gamma \setminus \Lambda$, if $X$ was an eigenvector of $\gamma$ with associated eigenvalue 1 we would have that $\gamma$ preserves the hyperplane $H$, thus any straight path from a point $P \in C \cap H$ to $\gamma P$ would descend to a loop in $N$ with holonomy $\gamma$, which is not in $\Lambda$, the image of $\pi_1 N$ by the holonomy.

## APPENDIX B. THE PROJECTIVE APPROACH TO AFFINE DEFORMATIONS

This appendix offers a more projective approach to affine deformations in order to link the work of this article to the notion of convex cocompact representation [DGK24].

Let $\Gamma$ be a torsion-free discrete subgroup of $SL(\mathbb{R}^{d+1})$ dividing an open proper convex cone $C$ in $\mathbb{R}^{d+1}$. Seeing the affine space $\mathbb{R}^{d+1}$ as an affine chart of the $(d+1)$-dimensional real projective space $\mathbb{P}(\mathbb{R}^{d+2})$:

$$ \left\{ \begin{array}{lcl} \mathbb{R}^{d+1} & \hookrightarrow & \mathbb{P}(\mathbb{R}^{d+2}) \\ X & \mapsto & (X:1), \end{array} \right. $$

we can see $SA(\mathbb{R}^{d+1}) = SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1}$ as a Lie subgroup of $PGL(\mathbb{R}^{d+2})$:

$$ \left\{ \begin{array}{lcl} SL(\mathbb{R}^{d+1}) \times \mathbb{R}^{d+1} & \hookrightarrow & PGL(\mathbb{R}^{d+2}) \\ (\gamma, \tau) & \mapsto & \begin{bmatrix} \gamma & \tau \\ 0 & 1 \end{bmatrix}. \end{array} \right. $$

As $\Gamma$ divides the cone $C = \{t(x, 1) | x \in \Omega, t > 0\}$, the domain $\Omega_P$ defined by

$$ \Omega_P := \{(x:1:0) | x \in \Omega\} \subseteq \mathbb{P}(\mathbb{R}^{d+2}) $$

is preserved by $\Gamma$ seen a subgroup of $PGL(\mathbb{R}^{d+2})$. Let $C_P$ be the cone over $\Omega_P$ with vertex at $[e_{d+2}] = (0: \cdots: 0:1) \in \mathbb{P}(\mathbb{R}^{d+2})$, i.e.

$$ C_P := \Omega_P \cup \{(tx: t: 1) | x \in \Omega, t \neq 0\} = \Omega_P \cup \{(X: 1) | X \in C\}. $$

Then $C_P$ is also preserved by $\Gamma$ and moreover, noticing that

$$ C_P = \{(x:1:\lambda) | x \in \Omega, \lambda \in \mathbb{R}\}, $$

we easily see that $C_P$ is convex and homeomorphic to $\Omega \times \mathbb{R}$. Its polar cone in the dual projective space $\mathbb{P}(\mathbb{R}^{d+2})^*$, identified with $\mathbb{P}(\mathbb{R}^{d+2})$ using the usual scalar product on $\mathbb{R}^{d+2}$, is homeomorphic to $\Omega^* \times \mathbb{R}$ and can be expressed as

$$ (B.1) \qquad C_P^* = \{(y: -1: \mu) | y \in \Omega^*, \mu \in \mathbb{R}\}. $$

Let $\Gamma_\tau$ be an affine deformation of $\Gamma$. An element $(\gamma, \tau_\gamma) \in \Gamma_\tau$ acts:

* on $\mathbb{P}(\mathbb{R}^{d+2})$ by

$$ (\gamma, \tau_\gamma) \cdot (X:1) = (\gamma X + \tau_\gamma: 1) \quad \text{and} \quad (\gamma, \tau_\gamma) \cdot (X:0) = (\gamma X: 0), $$

* on $\mathbb{P}(\mathbb{R}^{d+2})^*$ by

$$ (\gamma, \tau_\gamma) \star (Y:1) = (\gamma \star Y: 1 + \tau_{\gamma^{-1}} \star Y) \quad \text{and} \quad (\gamma, \tau_\gamma) \star (Y:0) = (\gamma \star Y: 0), $$

where $\gamma \star Y = \gamma^\top Y$. That dual action comes from the action of $SA^*(\mathbb{A}^{d+1})$, the group of *coaffine transformations* [BF24] seen as a subgroup of $PGL^*(\mathbb{R}^{d+1})$:

$$ \left\{ \begin{array}{lcl} SL^*(\mathbb{R}^{d+1}) \times (\mathbb{R}^{d+1})^* & \hookrightarrow & PGL^*(\mathbb{R}^{d+2}) \\ (\eta, \theta) & \mapsto & \begin{bmatrix} \eta & 0 \\ \theta & 1 \end{bmatrix}, \end{array} \right. $$
in which $\Gamma_{\tau}$ is seen as

$$ \left\{ \begin{bmatrix} \gamma^{-\top} & 0 \\ (\tau_{\gamma^{-1}})^{\top} & 1 \end{bmatrix} \in \mathrm{PGL}^*(\mathbb{R}^{d+1}) \mid \gamma \in \Gamma \right\}. $$

If $s$ is the support function of a $\tau$-convex domain, using the identification (B.1), we can see its graph as embedded in $\mathbb{P}(\mathbb{R}^{d+2})^*$:

$$ \mathrm{graph}_{\mathbb{P}}(s) := \{(y : -1 : -s(y)) \mid y \in \Omega^*\} = \{(Y : -\bar{s}(Y)) \mid Y \in \mathcal{C}^*\} $$

Then the $\tau$-equivariance (5.3) of $s$ precisely tells that $\mathrm{graph}_{\mathbb{P}}(s)$ is $\Gamma_{\tau}$-invariant. The existence of the unique boundary function $g_{\tau} : \partial\Omega^* \to \mathbb{R}$ (Proposition 5.12) thus tells us that there exist a $\Gamma_{\tau}$-invariant limit set $\Lambda_{\tau} \subset \partial\mathcal{C}_{\mathbb{P}}^*$ defined by

$$ \Lambda_{\tau}^{*} := \mathrm{graph}_{\mathbb{P}}(g_{\tau}) := \{(y : -1 : -g_{\tau}(y)) \mid y \in \Omega^{*}\}. $$

**Definition B.1** ([DGK24, Definition 1.1]). Let $G < \mathrm{PGL}(\mathbb{R}^{d+2})$ be an infinite discrete subgroup.

* Let $W$ be a $G$-invariant properly convex open subset of $\mathbb{P}(\mathbb{R}^{d+2})$. The action of $G$ on $W$ is strongly convex cocompact if $W$ is strictly convex with $C^1$ boundary and that for some $p \in W$, the convex hull in $W$ of the orbital limit set $\overline{\Gamma_{\tau}} \cdot p \cap \partial W$ is non-empty and has compact quotient by $G$.

* The group $G$ is strongly convex cocompact in $\mathbb{P}(\mathbb{R}^{d+2})$ if it admits a strongly convex cocompact action on some properly convex open subset $W$ of $\mathbb{P}(\mathbb{R}^{d+2})$.

**Proposition B.2.** When $\Gamma$ is hyperbolic, $\Gamma_{\tau}$ is strongly convex cocompact in $\mathbb{P}(\mathbb{R}^{d+2})^*$, and $\Lambda_{\tau}^*$ is its orbital limit set.

*Remark B.3.* By the work of Benoist [Ben04, Theorem 1.1], if $\Gamma$ is hyperbolic, then the domain $\Omega^*$ is strictly convex and $\partial\Omega^*$ is $C^1$.

*Proof.* For the fact that $\Gamma_{\tau}$ is strongly convex cocompact, one way to proceed is to realise that $\Gamma_{\tau} \simeq \Gamma$ is hyperbolic and show that the action of $\Gamma_{\tau}$ is projective-Anosov, as in [BF24, Lemma 3.6], and then use Theorem 1.4 from [DGK24]. In fact, we can explicitly build a $C^1$ strictly convex $\Gamma_{\tau}$-invariant domain of $\mathbb{P}(\mathbb{R}^{d+2})^*$ on which the action of $\Gamma_{\tau}$ is strongly convex cocompact. In Proposition 5.11 we proved the existence of $s_{-}$ and $s_{+}$, two smooth $\tau$-equivariant function on $\Omega^*$ which are respectively locally uniformly convex and locally uniformly concave satisfying that for all $y \in \partial\Omega^*$,

$$ (B.2) \qquad |\mathrm{grad}\,s_{\pm}(y')| \xrightarrow[y' \to y]{} +\infty. $$

Consider the following convex domain in $\mathbb{P}(\mathbb{R}^{d+2})^*$:

$$ W^* := \{(y : -1 : -\mu) \mid y \in \Omega^*, s_{-}(y) < \mu < s_{+}(y)\} \subset \mathcal{C}_{\mathbb{P}}^*. $$

We claim that the dual action of $\Gamma_{\tau}$ on $W^*$ is strongly convex cocompact.

We decompose the boundary of $W^*$ as

$$ \partial W^* = \mathrm{graph}_{\mathbb{P}}(s_{-}) \cup \Lambda_{\tau}^* \cup \mathrm{graph}_{\mathbb{P}}(s_{-}). $$

The strict convexity of $\Omega^*$ and the respective strict convexity and concavity of $s_{-}$ and $s_{+}$ tells us that there is no line segment in $\partial W^*$, i.e. that $W^*$ is strictly convex.

By Proposition 3.14, in order to prove that $\partial W^*$ is $C^1$, we only have to prove that at each point of its boundary $W^*$ admits only one supporting hyperplane. Using the smoothness of $s_{-}$ and $s_{+}$, that is clear for points in $\mathrm{graph}_{\mathbb{P}}(s_{+}) \cup \mathrm{graph}_{\mathbb{P}}(s_{-})$. Let $P = (y : -1 : -g_{\tau}(y))$ be a point of $\Lambda_{\tau}^* \subset \partial W^*$, then condition (B.2) implies that any supporting hyperplane to $W^*$ at $P$ is a supporting hyperplane (at $P$) to the convex domain
$C_{\mathbb{P}}^* = \{(y : -1 : \mu) | y \in \Omega^*, \mu \in \mathbb{R}\}$ which has $C^1$ boundary because of Remark B.3, and is thus unique.
Hence, we have that $\partial W^*$ is $C^1$.

Let $P = (y : -1 : -\mu)$ be a point of $W^*$. Using the well-known dynamic of the action of $\Gamma$ on $\Omega^*$ [Ben04, Proposition 5.1] and the fact that any $\tau$-invariant function extends continuously to $g_{\tau}$, we get that the proximal limit set of the action of $\Gamma_{\tau}$ on $W^*$ is $\bar{\Gamma}_{\tau} \cdot \bar{P} \cap \partial W^* = \Lambda_{\tau}^*$.

Finally, taking barycentres of its two boundary graphs, one can easily foliate $CH_{\mathbb{P}}^*(\tau)$ the convex hull of $\Lambda_{\tau}^*$ by $\Gamma_{\tau}$-invariant graphs. That means that it is cocompact for the action of $\Gamma_{\tau}$: it is the universal cover of the convex core $CC_{\mathbb{P}}^*(\tau) := CH_{\mathbb{P}}^*(\tau)/\Gamma_{\tau} \simeq \Omega^*/\Gamma \times [0, 1]$. $\square$

*Remark B.4.* Notice that when $\Gamma$ is not hyperbolic, we still have a limit set $\Lambda_{\tau}^*$, but it does not come from a projective-Anosov property: there is no limit map $\partial\Gamma_{\tau} \to \Lambda_{\tau}^*$.

## REFERENCES

[ABB$^+07] Lars Andersson, Thierry Barbot, Riccardo Benedetti, Francesco Bonsante, William M. Goldman, François Labourie, Kevin P. Scannell, and Jean-Marc Schlenker. Notes on a paper of Mess. *Geom. Dedicata*, 126:47–70, 2007.

[Abl] Antoine Ablondi. The equivariant affine Minkowski problem. In preparation.

[AGH98] Lars Andersson, Gregory J. Galloway, and Ralph Howard. The cosmological time function. *Classical Quantum Gravity*, 15(2):309–322, 1998.

[Bar05] Thierry Barbot. Globally hyperbolic flat space–times. *J. Geom. Phys.*, 53(2):123–165, 2005.

[Ben03] Yves Benoist. Divisible convex sets. II. *Duke Math. J.*, 120(1):97–120, 2003.

[Ben04] Yves Benoist. Divisible convex subsets. I. In *Algebraic groups and arithmetic. Proceedings of the international conference, Mumbai, India, December 17–22, 2001*, pages 339–374. New Delhi: Narosa Publishing House/Published for the Tata Institute of Fundamental Research, 2004.

[Ben05] Yves Benoist. Divisible convex sets. III. (Convexes divisibles. III). *Ann. Sci. Éc. Norm. Supér. (4)*, 38(5):793–832, 2005.

[Ben06] Yves Benoist. Divisible convex sets. IV: Boundary structure in dimension 3. *Invent. Math.*, 164(2):249–278, 2006.

[Ben08] Yves Benoist. A survey on divisible convex sets. In *Geometry, analysis and topology of discrete groups. Selected papers of the conference on geometry, topology and analysis of locally symmetric spaces and discrete groups*, Beijing, China, July 17–August 4, 2006, pages 1–18. Somerville, MA: International Press; Beijing: Higher Education Press, 2008.

[BF17] Francesco Bonsante and François Fillastre. The equivariant Minkowski problem in Minkowski space. *Ann. Inst. Fourier*, 67(3):1035–1113, 2017.

[BF20] Thierry Barbot and François Fillastre. Quasi-Fuchsian co-Minkowski manifolds. In *In the tradition of Thurston. Geometry and topology*, pages 645–703. Cham: Springer, 2020.

[BF24] M. D. Bobb and James Farre. Affine laminations and coaffine representations. Preprint, arXiv:2404.14284, 2024.

[Bon05] Francesco Bonsante. Flat spacetimes with compact hyperbolic Cauchy surfaces. *J. Differ. Geom.*, 69(3):441–521, 2005.

[BS16] Francesco Bonsante and Andrea Seppi. On Codazzi tensors on a hyperbolic surface and flat Lorentzian geometry. *Int. Math. Res. Not.*, 2016(2):343–417, 2016.

[Bur23] Guillaume Buro. *Les géodésiques des métriques finslériennes et pseudo-finslériennes faibles de basse régularité*. PhD thesis, EPFL, Lausanne, 2023.

[Bus67] Herbert Busemann. Timelike spaces. *Diss. Math.*, 53:1–50, 1967.

[BV24] Pierre-Louis Blayac and Gabriele Viaggi. Divisible convex sets with properly embedded cones. *Publ. Math. IHÉS*, pages 99–189, 2024.

[CG93] Suhyoung Choi and William M. Goldman. Convex real projective structures on closed surfaces are closed. *Proc. Am. Math. Soc.*, 118(2):657–661, 1993.

[Cho17] Suhyoung Choi. A classification of radial or totally geodesic ends of real projective orbifolds. I: A survey of results. In *Hyperbolic geometry and geometric group theory. Proceedings of the 7th Seasonal Institute of the Mathematical Society of Japan (MSJ-SI), Tokyo, Japan, July 30 – August 5, 2014*, pages 69–134. Tokyo: Mathematical Society of Japan (MSJ), 2017.

[Cho25] Suhyoung Choi. Real projective orbifolds with ends and their deformation theory: The deformation theory for nicest ones. Preprint, arXiv:2503.00438, 2025.

[CY77] Shiu-Yuen Cheng and Shing-Tung Yau. On the regularity of the Monge-Ampère equation $\det(\partial^2 u/\partial x_1 \partial x_j) = F(x, u)$. *Commun. Pure Appl. Math.*, 30:41–68, 1977.

[CY86] Shiu-Yuen Cheng and Shing-Tung Yau. Complete affine hypersurfaces. I: The completeness of affine metrics. *Commun. Pure Appl. Math.*, 39:839–866, 1986.

[Dan13] Jeffrey Danciger. A geometric transition from hyperbolic to anti-de Sitter geometry. *Geom. Topol.*, 17(5):3077–3134, 2013.

[DGK24] Jeffrey Danciger, François Guéritaud, and Fanny Kassel. Convex cocompact actions in real projective geometry. *Ann. Sci. Éc. Norm. Supér. (4)*, 57(6):1753–1843, 2024.
[Fil13] François Fillastre. Fuchsian convex bodies: basics of Brunn–Minkowski theory. *Geom. Funct. Anal.*, 23(1):295–333, 2013.

[FS19] François Fillastre and Andrea Seppi. Spherical, hyperbolic, and other projective geometries: convexity, duality, transitions. In *Eighteen essays in non-Euclidean geometry*, pages 321–409. Zürich: European Mathematical Society (EMS), 2019.

[FV16] François Fillastre and Giona Veronelli. Lorentzian area measures and the Christoffel problem. *Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5)*, 16(2):383–467, 2016.

[Ger70] Robert Geroch. Domain of dependence. *J. Math. Phys.*, 11:437–449, 1970.

[Gig81] Salvador Gigena. On a conjecture by E. Calabi. *Geom. Dedicata*, 11:387–396, 1981.

[Gol22] William M. Goldman. *Geometric structures on manifolds*, volume 227 of *Grad. Stud. Math*. Providence, RI: American Mathematical Society (AMS), 2022.

[Hit92] Nigel J. Hitchin. Lie groups and Teichmüller space. *Topology*, 31(3):449–473, 1992.

[JM87] Dennis Johnson and John J. Millson. Deformation spaces associated to compact hyperbolic manifolds. Discrete groups in geometry and analysis, Pap. Hon. G. D. Mostow 60th Birthday, Prog. Math. 67, 48–106 (1987)., 1987.

[Kos62] Jean-Louis Koszul. Ouverts convexes homogenes des espaces affines. *Math. Z.*, 79:254–259, 1962.

[Kos68] Jean-Louis Koszul. Déformations de connexions localement plates. *Ann. Inst. Fourier*, 18(1):103–114, 1968.

[KS18] Michael Kunzinger and Clemens Sämann. Lorentzian length spaces. *Ann. Global Anal. Geom.*, 54(3):399–447, 2018.

[Lab07] François Labourie. Flat projective structures on surfaces and cubic holomorphic differentials. *Pure Appl. Math. Q.*, 3(4):1057–1099, 2007.

[Lof01] John Loftin. Affine spheres and convex $\mathbb{RP}^n$-manifolds. *Am. J. Math.*, 123(2):255–274, 2001.

[Lof10] John Loftin. Survey on affine spheres. In *Handbook of geometric analysis. No. 2*, pages 161–191. Somerville, MA: International Press; Beijing: Higher Education Press, 2010.

[LSZH15] An-Min Li, Udo Simon, Guosong Zhao, and Zejun Hu. *Global affine differential geometry of hypersurfaces*, volume 11 of *De Gruyter Expo. Math*. Berlin: De Gruyter, 2nd revised and extended ed. edition, 2015.

[Mes07] Geoffrey Mess. Lorentz spacetimes of constant curvature. *Geom. Dedicata*, 126:3–45, 2007.

[Min17] Ettore Minguzzi. Affine sphere relativity. *Commun. Math. Phys.*, 350(2):749–801, 2017.

[Min19] Ettore Minguzzi. Causality theory for closed cone structures with applications. *Rev. Math. Phys.*, 31(5):139, 2019. Id/No 1930001.

[NS94] Katsumi Nomizu and Takeshi Sasaki. *Affine differential geometry*, volume 111 of *Camb. Tracts Math*. Cambridge: Cambridge University Press, 1994.

[NS22] Xin Nie and Andrea Seppi. Regular domains and surfaces of constant Gaussian curvature in 3-dimensional affine space. *Anal. PDE*, 15(3):643–697, 2022.

[NS23] Xin Nie and Andrea Seppi. Affine deformations of quasi-divisible convex cones. *Proc. Lond. Math. Soc. (3)*, 127(1):35–83, 2023.

[PY19] Athanase Papadopoulos and Sumio Yamada. Timelike Hilbert and Funk geometries. *Differ. Geom. Appl.*, 67:42, 2019. Id/No 101554.

[Roc97] R. Tyrrell Rockafellar. *Convex analysis*. Princeton, NJ: Princeton University Press, 1997.

[Sch93] Rolf Schneider. *Convex bodies: the Brunn-Minkowski theory*, volume 44 of *Encycl. Math. Appl*. Cambridge: Cambridge University Press, 1993.

[TW02] Neil S. Trudinger and Xu-Jia Wang. Affine complete locally convex hypersurfaces. *Invent. Math.*, 150(1):45–60, 2002.

[Vin63] E. B. Vinberg. The theory of convex homogeneous cones. *Trans. Mosc. Math. Soc.*, 12:340–403, 1963.

ANTOINE ABLONDI: IMAG, UNIVERSITÉ DE MONTPELLIER, CNRS, MONTPELLIER, FRANCE.

Email address: antoine.ablondi@umontpellier.fr