icci flow does not preserve positive sectional curvature in dimension fou

We find examples of cohomogeneity one metrics on $S^4$ and ${\mathds{C}}P^2$ with positive sectional curvature that lose this property when evolved via Ricci flow. These metrics are arbitrarily small perturbations of GroveZiller metrics with flat planes that become instantly negatively curved under Ricci flow.

Introduction
============

The Ricci flow $\tfrac{\partial}{\partial t}{\mathrm g}(t)=-2{\operatorname{Ric}}_{{\mathrm g}(t)}$ of Riemannian metrics on a smooth manifold is an evolution equation that continues to drive a wide range of breakthroughs in Geometric Analysis, see e.g. @bamler-survey for a survey. One of the keys to using Ricci flow is to control how the curvature of ${\mathrm g}(t)$ evolves; in particular, which curvature conditions of the original metric ${\mathrm g}(0)$ are preserved. Our main result establishes that, in dimension $n=4$, positive sectional curvature ($\sec>0$) is *not* among them:

\[mainthmA\] There exist smooth Riemannian metrics with $\sec>0$ on $S^4$ and ${\mathds{C}}P^2$ that evolve under the Ricci flow to metrics with sectional curvatures of mixed sign.

In contrast, $\sec>0$ is preserved on closed manifolds of dimension $n\leq 3$, by the seminal work of Hamilton @hamilton-original. Moreover, it was previously known @maximo2 that ${\operatorname{Ric}}>0$ is not preserved in dimension $n=4$, even among Kähler metrics, but these examples do not have $\sec>0$. Although does not readily extend to all $n>4$, there are examples of homogeneous metrics on flag manifolds of dimensions $6$, $12$, and $24$ with $\sec>0$ that lose that property when evolved via Ricci flow, see @bw-gafa-2007 [@cheung-wallach; @abiev-nikonorov]. A state-of-the-art discussion of Ricci flow invariant curvature conditions can be found in @bcrw, see also . builds on our earlier result @bettiol-krishnan1 that certain metrics with $\sec\geq0$, introduced by Grove and Ziller @grove-ziller-annals in a much broader context (see ), immediately acquire negatively curved planes on $S^4$ and ${\mathds{C}}P^2$, when evolved under Ricci flow. In light of the appropriate continuous dependence of Ricci flow on its initial data @BGI20, the metrics in are obtained by means of:

\[mainthmB\] Every Grove–Ziller metric on $S^4$ or ${\mathds{C}}P^2$ is the limit (in $C^\infty$-topology) of cohomogeneity one metrics with $\sec>0$.

In full generality, the problem of perturbing $\sec\geq0$ to $\sec>0$ is notoriously difficult, see e.g. @wilking-survey [Prob. 2]. Aside from clearly being unobstructed on $S^4$ and ${\mathds{C}}P^2$, the deformation problem is facilitated here by the presence of natural directions for perturbation, given by the round metric and the Fubini–Study metric, respectively. Indeed, we deform $\sec\geq0$ into $\sec>0$ in by linearly interpolating lengths of Killing vector fields for the ${\mathsf{SO}}(3)$-action which is isometric for both the Grove–Ziller metric ${\mathrm g}_0$ and the standard metric ${\mathrm g}_1$ on these spaces. The resulting ${\mathsf{SO}}(3)$-invariant metrics ${\mathrm g}_s$, $s\in [0,1]$, are smooth and have $\sec>0$ for all sufficiently small $s>0$. For a lower-dimensional illustration, consider the ${\mathsf{T}}^2$-action on $S^3\subset {\mathds{C}}^2$ via $(e^{i\theta_1},e^{i\theta_2})\cdot(z,w)=\big(e^{i\theta_1}z,e^{i\theta_2}w\big)$, and invariant metrics $${\mathrm g}={\mathrm{d}}r^2+ \varphi(r)^2\,{\mathrm{d}}\theta_1^2+\xi(r)^2\,{\mathrm{d}}\theta_2^2, \quad 0<r<\tfrac\pi2,$$ written along the geodesic segment $\gamma(r)=(\sin r,\cos r)$. The functions $\varphi$ and $\xi$ encode the ${\mathrm g}$-lengths of the Killing fields $\frac{\partial}{\partial \theta_1}$ and $\frac{\partial}{\partial \theta_2}$ respectively, and must satisfy certain smoothness conditions at the endpoints $r=0$ and $r=\frac\pi2$. The unit round metric ${\mathrm g}_1$ is given by setting $\varphi$ and $\xi$ to be $\varphi_1(r)=\sin r$ and $\xi_1(r)=\cos r$, while a Grove–Ziller metric ${\mathrm g}_0$ corresponds to concave monotone functions $\varphi_0$ and $\xi_0$ that plateau at a constant value $b>0$ for at least half of $\left[0,\frac\pi2\right]$. The curvature operator of ${\mathrm g}$ is easily seen to be diagonal, with eigenvalues $-\varphi''/\varphi$, $-\xi''/\xi$, and $-\varphi'\xi'/\varphi\xi$, see e.g. @petersen-book-3 [Sec. 4.2.4], so it has $\sec\geq0$ if and only if $\varphi$ and $\xi$ are concave and monotone, and $\sec>0$ if and only if they are *strictly* concave and monotone. Thus, $$\varphi_s=(1-s)\,\varphi_0+s\,\varphi_1 \quad \text{ and }\quad \xi_s=(1-s)\,\xi_0+s\,\xi_1$$ give rise to metrics ${\mathrm g}_s$ deforming ${\mathrm g}_0$ to have $\sec>0$ for $s>0$. It turns out that a similar approach works for proving , with the addition of a third (nowhere vanishing) function $\psi$, to deal with ${\mathsf{SO}}(3)$-invariant metrics on $4$-manifolds. The biggest challenge is verifying that these metrics have $\sec>0$, since that is no longer equivalent to positive-definiteness of the curvature operator if $n\geq4$. To overcome this difficulty, we use a much simpler algebraic characterization of $\sec>0$ in dimension $n=4$, given by the Finsler–Thorpe trick ().

Motivated by the above, it is natural to ask whether the set of cohomogeneity one metrics with $\sec\geq0$ on a given closed manifold coincides with the closure (say, in $C^2$-topology) of the set of such metrics with $\sec>0$, if the latter is nonempty. In contrast to , there is some evidence to suggest that Grove–Ziller metrics on certain $7$-manifolds cannot be perturbed to have $\sec>0$, see @ziller-coh1survey [Sec. 4].

This paper is organized as follows. Background material on cohomogeneity one manifolds and the Finsler–Thorpe trick in dimension $4$ is presented in . The smoothness conditions and curvature operator of ${\mathsf{SO}}(3)$-invariant metrics on $S^4$ and ${\mathds{C}}P^2$ are discussed in . contains the proof of , focusing mainly on the case of $S^4$, since the proof for ${\mathds{C}}P^2$ is mostly analogous. Finally, is proved in .

Preliminaries {#sec:prelim}
=============

Cohomogeneity one
-----------------

We briefly discuss the geometry of cohomogeneity one manifolds to fix notations, see @mybook [@bettiol-krishnan1; @grove-ziller-annals; @gz-inventiones; @VZ18; @ziller-coh1survey] for details.

A cohomogeneity one manifold is a Riemannian manifold $(M,{\mathrm g})$ endowed with an isometric action by a Lie group ${\mathsf{G}}$, such that the orbit space $M/{\mathsf{G}}$ is one-dimensional. Let $\pi\colon M\to M/{\mathsf{G}}$ be the projection map. Throughout, we assume $M/{\mathsf{G}}=[0,L]$ is a closed interval, and the nonprincipal orbits $B_-=\pi^{-1}(0)$ and $B_+=\pi^{-1}(L)$ are *singular orbits*. In other words, $B_\pm$ are smooth submanifolds of dimension strictly smaller than the principal orbits $\pi^{-1}(r)$, $r\in (0,L)$, which are smooth hypersurfaces of $M$. Fix $x_{-}\in B_{-}$, and consider a minimal geodesic $\gamma(r)$ in $M$ joining $x_{-}$ to $B_{+}$, meeting it at $x_{+}=\gamma(L)$; that is, $\gamma$ is a horizontal lift of $[0,L]$ to $M$. Denote by ${\mathsf{K}}_{\pm}$ the isotropy group at $x_{\pm}$, and by ${\mathsf{H}}$ the isotropy at $\gamma(r)$, for $r\in (0,L)$. By the Slice Theorem, given $r_{\mathrm{max}}^\pm>0$ so that $r_{\mathrm{max}}^+ +r_{\mathrm{max}}^-=L$, the tubular neighborhoods $D(B_{-}) = \pi^{-1}\left(\left[0, r_{\mathrm{max}}^-\right]\right)$ and $D(B_{+}) = \pi^{-1}\left(\left[L-r_{\mathrm{max}}^+ ,L\right]\right)$ of the singular orbits are disk bundles over $B_-$ and $B_+$. Let $D^{l_{\pm}+1}$ be the normal disks to $B_{\pm}$ at $x_{\pm}$. Then ${\mathsf{K}}_{\pm}$ acts transitively on the boundary $\partial D^{l_{\pm}+1}$, with isotropy ${\mathsf{H}}$, so $\partial D^{l_{\pm}+1} = S^{l_{\pm}} = {\mathsf{K}}_{\pm}/{\mathsf{H}}$, and the ${\mathsf{K}}_{\pm}$-action on $\partial D^{l_{\pm}+1}$ extends to a ${\mathsf{K}}_\pm$-action on all of $D^{l_{\pm}+1}$. Moreover, there are equivariant diffeomorphisms $D(B_{\pm}) \cong{\mathsf{G}}\times_{{\mathsf{K}}_{\pm}}D^{l_{\pm}+1}$, and $M\cong D(B_-)\cup D(B_+)$, where the latter is given by gluing these disk bundles along their common boundary $\partial D(B_{\pm}) \cong{\mathsf{G}}\times_{{\mathsf{K}}_{\pm}} S^{l_{\pm}}\cong {\mathsf{G}}/{\mathsf{H}}$. In this situation, one associates to $M$ the *group diagram* $${\mathsf{H}}\subset\{{\mathsf{K}}_-,{\mathsf{K}}_+\}\subset {\mathsf{G}}.$$ Conversely, given a group diagram as above, where ${\mathsf{K}}_{\pm}/{\mathsf{H}}$ are spheres, there exists a cohomogeneity one manifold $M$ given as the union of the above disk bundles.

Fix a bi-invariant metric $Q$ on the Lie algebra ${\mathfrak{g}}$ of ${\mathsf{G}}$, and set ${\mathfrak{n}}= {\mathfrak{h}}^\perp$, where ${\mathfrak{h}}\subset{\mathfrak{g}}$ is the Lie algebra of ${\mathsf{H}}$. Identifying ${\mathfrak{n}}\cong T_{\gamma(r)}({\mathsf{G}}/{\mathsf{H}})$ for each $0<r<L$ via action fields $X\mapsto X^*_{\gamma(r)}$, any ${\mathsf{G}}$-invariant metric on $M$ can be written as $$\label{eqn:coh1metric}
 {\mathrm g}={\mathrm{d}}r^2 + {\mathrm g}_r, \quad  0<r<L,$$ along the geodesic $\gamma(r)$, where ${\mathrm g}_r$ is a $1$-parameter family of left-invariant metrics on ${\mathsf{G}}/{\mathsf{H}}$, i.e., of ${\operatorname{Ad}}({\mathsf{H}})$-invariant metrics on ${\mathfrak{n}}$. As $r\searrow0$ and $r\nearrow L$, the metrics ${\mathrm g}_r$ degenerate, according to how ${\mathsf{G}}(\gamma(r))\cong{\mathsf{G}}/{\mathsf{H}}$ collapse to $B_\pm={\mathsf{G}}/{\mathsf{K}}_\pm$. Namely, they satisfy *smoothness conditions* that characterize when a tensor defined by means of on $M\setminus (B_-\cup B_+)\cong (0,L)\times {\mathsf{G}}/{\mathsf{H}}$ extends smoothly to all of $M$, see @VZ18.

### Grove–Ziller metrics {#subsec:GZ-metrics}

If both singular orbits $B_\pm$ of a cohomogeneity one manifold $M$ have codimension two, then $M$ can be endowed with a new ${\mathsf{G}}$-invariant metric ${\mathrm g_{\mathrm{GZ}}}$ with $\sec\geq0$, as shown in the celebrated work of Grove and Ziller @grove-ziller-annals [Thm. 2.6]. We now describe this construction, building metrics with $\sec\geq0$ on each disk bundle $D(B_\pm)$ that restrict to a fixed product metric ${\mathrm{d}}r^2+b^2 Q|_{\mathfrak n}$ near $\partial D(B_\pm)\cong {\mathsf{G}}/{\mathsf{H}}$, so that these two pieces can be isometrically glued together.

Consider one such disk bundle $D(B)$ at a time, say over a singular orbit $B={\mathsf{G}}/{\mathsf{K}}$, and let ${\mathfrak{k}}$ be the Lie algebra of ${\mathsf{K}}$. Set ${\mathfrak{m}}= {\mathfrak{k}}^\perp$ and ${\mathfrak{p}}= {\mathfrak{h}}^\perp \cap {\mathfrak{k}}$, so that ${\mathfrak{g}}={\mathfrak{m}}\oplus{\mathfrak{p}}\oplus{\mathfrak{h}}$ is a $Q$-orthogonal direct sum. Since ${\mathfrak{p}}$ is $1$-dimensional, the metric $Q_{a,b}$ on ${\mathsf{G}}$, given by $$\begin{aligned}
  Q_{a,b}|_{\mathfrak{m}}:= b^2\,  Q|_{\mathfrak{m}}, \qquad Q_{a,b}|_{\mathfrak{p}}:= ab^2\, Q|_{\mathfrak{p}}, \qquad Q_{a,b}|_{\mathfrak{h}}:=b^2\, Q|_{\mathfrak{h}},\end{aligned}$$ has $\sec \geq 0$ whenever $0<a \leq \frac{4}{3}$ and $b>0$, see @grove-ziller-annals [Prop. 2.4] or @bm-mathann [Lemma 3.2]. Fix $1<a\leq \frac43$, and let $r_{\mathrm{max}}>0$ be such that $$\label{eq:MVT-obstruction}
y:=\tfrac{ \rho\sqrt{a}}{\sqrt{a-1}} \;\;\text{ satisfies }\;\; y< \, r_{\mathrm{max}},$$ where $\rho=\rho(b)$ is the radius of the circle(s) ${\mathsf{K}}/{\mathsf{H}}$ endowed with the metric $b^2\,Q|_{{\mathfrak{p}}}$. Then, we can find a smooth nondecreasing function $f\colon \left[0,r_{\mathrm{max}}\right]\to{\mathds{R}}$ and some $0<r_0<r_{\mathrm{max}}$, with $f(0)=0$, $f'(0)=1$, $f^{(2n)}(0)=0$ for all $n\in\mathds N$, $f''(r)\leq 0$ for all $r\in \left[0,r_{\mathrm{max}}\right]$, $f^{(3)}(r) > 0$ for all $r\in [0, r_0)$, and $f(r) \equiv y$ for all $r\in \left[r_0, r_{\mathrm{max}}\right]$. The rotationally symmetric metric ${\mathrm g}_{D^2} = {\mathrm{d}}r^2 + f(r)^2 {\mathrm{d}}\theta^2$, $0<r\leq r_{\mathrm{max}}$, on the punctured disk $D^2\setminus\{0\}$ extends to a smooth metric ${\mathrm g}_{D^2}$ on $D^2$ with $\sec\geq0$ that, near $\partial D^2=\{r=r_{\mathrm{max}}\}$, is isometric to a round cylinder $\left[r_0, r_{\mathrm{max}}\right]\times S^1(y)$ of radius $y$. Thus, the product manifold $({\mathsf{G}}\times D^2, Q_{a,b} + {\mathrm g}_{D^2})$ has $\sec \geq 0$, and so does the orbit space $D(B)\cong {\mathsf{G}}\times_{\mathsf{K}}D^2$ of the ${\mathsf{K}}$-action on ${\mathsf{G}}\times D^2$, when endowed with the metric ${\mathrm g_{\mathrm{GZ}}}$ that makes the projection map $\Pi\colon ({\mathsf{G}}\times D^2, Q_{a,b} + {\mathrm g}_{D^2})\to ({\mathsf{G}}\times_{\mathsf{K}}D^2,{\mathrm g_{\mathrm{GZ}}})$ a Riemannian submersion. Writing this metric ${\mathrm g_{\mathrm{GZ}}}$ in the form , we have $$\label{eq:gGZcoh1form}
    {\mathrm g_{\mathrm{GZ}}}={\mathrm{d}}r^2+b^2\, Q|_{{\mathfrak{m}}} +\tfrac{f(r)^2a}{f(r)^2 + a \rho^2}b^2\,Q|_{{\mathfrak{p}}}, \quad 0<r\leq r_{\mathrm{max}},$$ see e.g. @grove-ziller-annals [Lemma 2.1, Rem. 2.7] or @bm-mathann [Lemma 3.1 (ii)]. In particular, ${\mathrm g_{\mathrm{GZ}}}={\mathrm{d}}r^2 +b^2\, Q|_{{\mathfrak{n}}}$ for all $r\in \left[r_0, r_{\mathrm{max}}\right]$, since $\tfrac{f(r)^2a}{f(r)^2 + a \rho^2}\equiv 1$ for all such $r$; hence $(D(B),{\mathrm g_{\mathrm{GZ}}})$ is isometric to the prescribed product metric near $\partial D(B)\cong {\mathsf{G}}/{\mathsf{H}}$.

This construction can be performed on each disk bundle $D(B_\pm)$ with the same $b>0$, provided $r_{\mathrm{max}}^\pm>0$ are chosen sufficiently large so that holds for the corresponding radii $\rho_\pm(b)$ of the circles ${\mathsf{K}}_\pm/{\mathsf{H}}$ endowed with the metric $b^2\,Q|_{\mathfrak p_\pm}$. Gluing these two disk bundles together, we obtain the desired ${\mathsf{G}}$-invariant metric ${\mathrm g_{\mathrm{GZ}}}$ with $\sec\geq0$ on $M\cong D(B_-)\cup D(B_+)$ and $M/{\mathsf{G}}=[0,L]$, where $L = r_{\mathrm{max}}^+ + r_{\mathrm{max}}^-$. Although it is natural to pick the same (largest) value for $r_{\mathrm{max}}^\pm$, so that the gluing occurs at $r=\frac{L}{2}$, it is convenient to not impose this restriction. Note that $$\label{eq:lowerboundL}
    L  = r_{\mathrm{max}}^+ + r_{\mathrm{max}}^- > 
    \tfrac{\sqrt{a}}{\sqrt{a-1}}\,\big(\rho_+(b) +\rho_-(b)\big),
        $$ if the gluing interface $\partial D(B_\pm)$ is isometric to $({\mathsf{G}}/{\mathsf{H}},b^2 Q|_{\mathfrak{n}})$. Conversely, given $1<a\leq\frac43$, $b>0$, and $L$ satisfying , there exists a Grove–Ziller metric on $M$ with gluing interface $({\mathsf{G}}/{\mathsf{H}},b^2 Q|_{\mathfrak{n}})$, induced by $Q_{a,b}+{\mathrm g}_{D^2}$, and with $M/{\mathsf{G}}=[0,L]$.

\[rem:psec0r1\]\] Although this is not a requirement in the original Grove–Ziller construction, we assume that $f^{(3)}(r) > 0$ on $[0, r_0)$, hence the curvature of $(D^2, {\mathrm g}_{D^2})$ is monotonically decreasing for $r\in [0,r_0)$. As a consequence, for each $0<r_*<r_0$, there is a constant $c>0$, depending on $r_*$, so that $\sec_{{\mathrm g}_{D^2}}  \geq c$ for all $r\in [0,r_*]$.

Finsler–Thorpe trick
--------------------

In order to verify $\sec>0$ on Riemannian $4$-manifolds, we shall use a result that became known in the Geometric Analysis community as *Thorpe's trick*, attributed to Thorpe @Thorpe72, but that actually follows from much earlier work of Finsler @finsler, and is often referred to as *Finsler's Lemma* in Convex Algebraic Geometry. This rather multifaceted result is also known as the *$S$-lemma*, or *$S$-procedure*, in the mathematical optimization and control literature, see e.g. @slemma-survey. Details and other geometric perspectives can be found in @bkm-siaga.

Let ${\operatorname{Sym}}^2_{\mathrm b}(\wedge^2{\mathds{R}}^n)\subset {\operatorname{Sym}}^2(\wedge^2{\mathds{R}}^n)$ be the subspace of symmetric endomorphisms $R\colon \wedge^2{\mathds{R}}^n\to\wedge^2{\mathds{R}}^n$ that satisfy the first Bianchi identity. These objects are called *algebraic curvature operators*, and serve as pointwise models for the curvature operators of Riemannian $n$-manifolds. For instance, $R\in{\operatorname{Sym}}^2_{\mathrm b}(\wedge^2{\mathds{R}}^n)$ is said to have $\sec\geq0$, respectively $\sec>0$, if the restriction of the quadratic form $\langle R(\sigma),\sigma\rangle$ to the oriented Grassmannian $\operatorname{Gr}_2^+({\mathds{R}}^n)\subset\wedge^2{\mathds{R}}^n$ of $2$-planes is nonnegative, respectively positive. A Riemannian manifold $(M^n,{\mathrm g})$ has $\sec\geq0$, or $\sec>0$, if and only if its curvature operator $R_p\in{\operatorname{Sym}}^2_{\mathrm b}(\wedge^2 T_pM)$ has $\sec\geq0$, or $\sec>0$, for all $p\in M$.

The orthogonal complement to ${\operatorname{Sym}}^2_{\mathrm b}(\wedge^2{\mathds{R}}^n)$ is identified with $\wedge^4{\mathds{R}}^n$; so, if $n=4$, it is $1$-dimensional, and spanned by the Hodge star operator $*$. Since $\sigma\in\wedge^2{\mathds{R}}^4$ satisfies $\sigma\wedge\sigma=0$ if and only if $\langle *\sigma,\sigma\rangle=0$, the quadric defined by $*$ in $\wedge^2{\mathds{R}}^4$ is precisely the Plücker embedding $\operatorname{Gr}_2^+({\mathds{R}}^4)\subset\wedge^2{\mathds{R}}^4$. As shown by Finsler @finsler, a quadratic form $\langle R(\sigma),\sigma\rangle$ is nonnegative when restricted to the quadric $\langle *\sigma,\sigma\rangle=0$ if and only if some linear combination of $R$ and $*$ is positive-semidefinite, yielding:

\[Finsler–Thorpe trick\]\[prop:FTtrick\] Let $R\in {\operatorname{Sym}}^2_{\mathrm b}(\wedge^2 {\mathds{R}}^4)$ be an algebraic curvature operator. Then $R$ has $\sec\geq0$, respectively $\sec>0$, if and only if there exists $\tau\in{\mathds{R}}$ such that $R+\tau\, *\succeq0$, respectively $R+\tau\, *\succ0$.

\[rem:set-of-taus\] For a given $R\in {\operatorname{Sym}}^2_{\mathrm b}(\wedge^2 {\mathds{R}}^4)$ with $\sec\geq0$, the set of $\tau\in{\mathds{R}}$ such that $R+\tau\,*\succeq0$ is a closed interval $[\tau_{\mathrm{min}},\tau_{\mathrm{max}}]$, which degenerates to a single point, i.e., $\tau_{\mathrm{min}}=\tau_{\mathrm{max}}$, if and only if $R$ does not have $\sec>0$, see @bkm-siaga [Prop. 3.1]

The equivalences given by Finsler–Thorpe's trick offer substantial computational advantages to test for $\sec\geq0$ or $\sec>0$, see the discussion in @bkm-siaga [Sec. 5.4].

Cohomogeneity one structure of the sphere and complex projective plane {#sec:s4andcp2}
======================================================================

Both $S^4$ and ${\mathds{C}}P^2$ admit a cohomogeneity one action by ${\mathsf{G}}={\mathsf{SO}}(3)$ as we now recall, see @bettiol-krishnan1 [Sec. 3] and @ziller-coh1survey [Sec. 2] for details. The ${\mathsf{G}}$-action on $S^4$ is the restriction to the unit sphere of the ${\mathsf{SO}}(3)$-action by conjugation on the space of symmetric traceless $3\times 3$ real matrices, while the ${\mathsf{G}}$-action on ${\mathds{C}}P^2$ is a subaction of the transitive ${\mathsf{SU}}(3)$-action. The corresponding orbit spaces are $S^4/{\mathsf{G}}=\left[0,\frac\pi3\right]$ and ${\mathds{C}}P^2/{\mathsf{G}}=\left[0,\frac\pi4\right]$, endowing $S^4$ with the round metric with $\sec\equiv1$, and ${\mathds{C}}P^2$ with the Fubini–Study metric with $1\leq\sec\leq4$. Their group diagrams are as follows: $$\begin{aligned}
   S^4&:  &{\mathds{Z}}_2\oplus{\mathds{Z}}_2\cong {\mathsf{S}}({\mathsf{O}}(1){\mathsf{O}}(1){\mathsf{O}}(1)) &\subset \{  {\mathsf{S}}({\mathsf{O}}(1){\mathsf{O}}(2)),{\mathsf{S}}({\mathsf{O}}(2){\mathsf{O}}(1))\} \subset{\mathsf{SO}}(3),\\
   {\mathds{C}}P^2&: &{\mathds{Z}}_2\cong \langle\operatorname{diag}(-1,-1,1) \rangle &\subset \{ {\mathsf{S}}({\mathsf{O}}(1){\mathsf{O}}(2)), {\mathsf{SO}}(2)_{1,2}\} \subset{\mathsf{SO}}(3),\end{aligned}$$ according to an appropriate choice of minimal geodesic $\gamma(r)$, $r\in [0,L]$, see @bettiol-krishnan1 [Sec. 3]. In both cases, since ${\mathsf{H}}$ is discrete, ${\mathfrak{n}}\cong{\mathfrak{g}}= \mathfrak{so}(3)$. We henceforth fix $Q$ to be the bi-invariant metric such that $\{E_{23}, E_{31}, E_{12}\}$ is a $Q$-orthonormal basis of $\mathfrak{so}(3)$, where $E_{ij}$ is the skew-symmetric $3\times 3$ matrix with a $+1$ in the $(i,j)$ entry, a $-1$ in the $(j,i)$ entry, and zeros in the remaining entries. The $1$-dimensional subspaces ${\mathfrak{n}}_k=\operatorname{span}(E_{ij})$, where $(i,j,k)$ is a cyclic permutation of $(1,2,3)$, are pairwise inequivalent for the adjoint action of ${\mathsf{H}}$ in the case of $S^4$, while ${\mathfrak{n}}_1$ and ${\mathfrak{n}}_2$ are equivalent in the case of ${\mathds{C}}P^2$, but neither is equivalent to ${\mathfrak{n}}_3$.

Collectively denoting $S^4$ and ${\mathds{C}}P^2$ with the above cohomogeneity one structures by $M^4$, we consider *diagonal* ${\mathsf{G}}$-invariant metrics ${\mathrm g}$ on $M^4$, i.e., metrics of the form $$\label{eq:g-phi,psi,xi}
 {\mathrm g}={\mathrm{d}}r^2+ \varphi(r)^2 \, Q|_{{\mathfrak{n}}_1} + \psi(r)^2 \, Q|_{{\mathfrak{n}}_2} + \xi(r)^2 \, Q|_{{\mathfrak{n}}_3}, \quad 0<r<L,$$ where $L=\frac\pi3$ or $L=\frac\pi4$ according to whether $M^4=S^4$ or $M^4={\mathds{C}}P^2$, cf. . Note that every ${\mathsf{G}}$-invariant metric on $S^4$ is of the above form, i.e., ${\mathfrak{n}}_k$ are pairwise orthogonal, but ${\mathfrak{n}}_1$ and ${\mathfrak{n}}_2$ need not be orthogonal for all ${\mathsf{G}}$-invariant metrics on ${\mathds{C}}P^2$, i.e., the off-diagonal term ${\mathrm g}(E_{23},E_{31})$ need not vanish identically. The standard metric on $M^4$, with curvatures normalized as above, is obtained setting $\varphi,\psi,\xi$ to $$\label{eq:can-phi,psi,xi}
\begin{aligned}
   S^4&:  &\varphi_1(r)&=2\sin r,  &\psi_1(r)&= \sqrt{3}\cos r + \sin r,  &\xi_1(r)&= \sqrt{3}\cos r - \sin r, \\
   {\mathds{C}}P^2&: &\varphi_1(r)&=\sin r,  &\psi_1(r)&= \cos r,  &\xi_1(r)&= \cos 2r,
\end{aligned}$$ see below for their graphs.

Smoothness
----------

The conditions required of $\varphi, \psi, \xi$ for the metric ${\mathrm g}$ in , which is defined on the open dense set $M^4\setminus (B_-\cup B_+)\cong (0,L)\times {\mathsf{G}}/{\mathsf{H}}$, to extend smoothly to all of $M^4$ can be extracted from @VZ18 [Sec. 3.1, 3.2] as follows:

\[prop:smoothness\] The ${\mathsf{G}}$-invariant metric on $M^4\setminus (B_-\cup B_+)$ extends to a smooth metric on $M^4$ if and only if $\varphi,\psi,\xi$ extend smoothly to $r=0$ and $r=L$ satisfying the following, where $\phi_k$ are smooth, $z=L-r$, and $\varepsilon>0$ is small:

<span>|c|l|</span> $M^4$ &

------------------------------------------------------------------------

------------------------------------------------------------------------

Smoothness conditions on $\varphi,\psi,\xi$\
$\begin{array}{c}
S^4 \\[5pt]
L =\frac\pi3
\end{array}$ & $\begin{array}{l}
          {\rm (i)} \; \varphi(0) = 0,\,  \rule{0pt}{2.5ex} \varphi'(0) = 2, \,\varphi^{(2n)}(0) = 0, \text{  for all } n \geq 1, \\[1pt] 
          {\rm (ii)} \;\psi(r)^2 + \xi(r)^2 = \phi_1(r^2), \text{  for all } r\in [0,\varepsilon), \\[1pt] 
          {\rm (iii)} \;\psi(r)^2 - \xi(r)^2 = r\,\phi_2(r^2), \text{  for all } r\in [0,\varepsilon), \\[3pt]
                     {\rm (iv)} \;\xi(L) = 0, \, \xi'(L) = -2, \, \xi^{(2n)}(L) = 0, \text{  for all } n \geq 1, \\[1pt]
          {\rm (v)} \;\psi(z)^2 + \varphi(z)^2 = \phi_3(z^2), \text{  for all } z\in [0,\varepsilon), \\[1pt] 
          {\rm (vi)} \;\psi(z)^2 - \varphi(z)^2 = z\,\phi_4(z^2), \text{  for all } z\in [0,\varepsilon).   $r(3-,3\]$;
        \end{array}$\
$\begin{array}{c}
{\mathds{C}}P^2 \\[5pt]
L =\frac\pi4
\end{array}$ & $\begin{array}{l}
         {\rm (i)} \; \varphi(0) = 0,\,  \rule{0pt}{2.5ex} \varphi'(0) = 1, \,\varphi^{(2n)}(0) = 0, \text{  for all } n \geq 1, \\[1pt] 
          {\rm (ii)} \;\psi(r)^2 + \xi(r)^2 = \phi_5(r^2), \text{  for all } r\in [0,\varepsilon), \\[1pt] 
         {\rm (iii)} \; \psi(r)^2 - \xi(r)^2 = r^2\,\phi_6(r^2), \text{  for all } r\in [0,\varepsilon), \\[3pt]
                     {\rm (iv)} \; \xi(L) = 0, \, \xi'(L) = -2, \, \xi^{(2n)}(L) = 0, \text{  for all } n \geq 1, \\[1pt]
         {\rm (v)} \;  \psi(z)^2 + \varphi(z)^2 = \phi_7(z^2), \text{  for all } z\in [0,\varepsilon), \\[1pt] 
         {\rm (vi)} \;  \psi(z)^2 - \varphi(z)^2 = z\,\phi_8(z^2), \text{  for all } z\in [0,\varepsilon). $r(3-,3\]$;
        \end{array}$\

\[rem:extra-symm\] Since the isotropy groups ${\mathsf{K}}_\pm$ for the ${\mathsf{G}}$-action on $S^4$ are conjugate, the smoothness conditions at the endpoints $r=0$ and $r=L$ can be obtained from one another by interchanging the roles of $\varphi$ and $\xi$. Furthermore, just as the round metric , all metrics we consider on $S^4$ have the following additional symmetries: $$\label{eqn:gS4sym}
 \varphi(r)=\xi\left(L-r\right), \quad \text{and}\quad \psi(r)=\psi\left(L-r\right), \quad \text{ for all } 0\leq r\leq L.$$ However, metrics on ${\mathds{C}}P^2$ do not have any of these features or extra symmetries, as ${\mathsf{K}}_\pm$ are not conjugate, and, in general $\varphi(r) \neq \xi\left(L-r\right)$ and $\psi(r)\neq\psi\left(L-r\right)$.

\[scale=0.75\]

\[ axis x line=middle, axis y line=middle, axis line style = thick, tick style = thick, ymax=2.1, xmax=<span>pi/3+.1</span>, xtick=<span>pi/6,pi/3</span>, xticklabels=<span>$\frac\pi6$, $\frac\pi3$</span>, ytick=<span>sqrt(3)</span>, yticklabels=<span>$\sqrt3$</span>, \] ; ; ; (pi/4,1.55) node <span>$\color{red}\varphi_1$</span>; (pi/4,2.05) node <span>$\color{black!40!green}\psi_1$</span>; (pi/4,0.69) node <span>$\color{blue}\xi_1$</span>;

\[scale=0.75\]

\[ axis x line=middle, axis y line=middle, axis line style = thick, tick style = thick, ymax=2.1, xmax=<span>pi/4+.1</span>, xtick=<span>pi/8,pi/6,pi/4</span>, xticklabels=<span>$\frac\pi8$, $\frac{\pi}{6}$, $\frac\pi4$</span>, ytick=<span>sqrt(2)/2, 1</span>, yticklabels=<span>$\frac{\sqrt2}{2}$, $1$</span>, \] ; ; ; (pi/5,0.67) node <span>$\color{red}\varphi_1$</span>; (pi/5,0.9) node <span>$\color{black!40!green}\psi_1$</span>; (pi/5,0.42) node <span>$\color{blue}\xi_1$</span>;

Curvature
---------

Computing the curvature operator of the ${\mathsf{G}}$-invariant metric on $M^4$, with the formulae in @gz-inventiones [Prop. 1.12], one obtains the following:

\[propn:curv\_op\] Let $\{ e_i \}_{i=0}^3$ be the ${\mathrm g}$-orthonormal frame along the geodesic $\gamma(r)$, $0<r<L$, given by $e_0=\gamma'(r)$, $e_1 = \frac{1}{\varphi(r)} E_{23}^*$, $e_2 = \frac{1}{\psi(r)} E_{31}^*$, $e_3 = \frac{1}{\xi(r)} E_{12}^*$, i.e., $e_0$ is the unit horizontal direction and $\{e_1,e_2,e_3\}$ are unit Killing vector fields. In the basis ${\mathcal{B}}:=\{ 
e_2\wedge e_3,\, e_0\wedge e_1,\, 
e_3\wedge e_1,\, e_0\wedge e_2,\,
e_1\wedge e_2,\, e_0\wedge e_3
\},$ the curvature operator ${R}\colon\wedge^2 T_{\gamma(r)}M^4 \to\wedge^2 T_{\gamma(r)}M^4$, $0<r<L$, is block diagonal, that is, ${R}= \operatorname{diag}({R}_1, {R}_2, {R}_3)$, with $2\times 2$ blocks given as follows: $$\begin{aligned}
 {R}_1 &= \begin{bmatrix}
  \frac{\psi^4+\xi^4 -\varphi^4 + 2(\xi^2-\varphi^2)(\varphi^2-\psi^2)}{4\varphi^2  \psi^2\xi^2 } - \frac{\psi'\xi'}{\psi\xi }   &
    \; \frac{\psi'(\psi^2+\varphi^2-\xi^2)}{2 \varphi\psi^2\xi} + \frac{\xi'(\xi^2+\varphi^2-\psi^2)}{2\varphi\psi\xi^2} -\frac{\varphi'}{\psi\xi}   \\[5pt]
   \; \frac{\psi'(\psi^2+\varphi^2-\xi^2)}{2 \varphi\psi^2\xi} + \frac{\xi'(\xi^2+\varphi^2-\psi^2)}{2\varphi\psi\xi^2} -\frac{\varphi'}{\psi\xi} &
    -\frac{\varphi''}{\varphi}
 \end{bmatrix},\\[3pt]
  {R}_2 &= \begin{bmatrix}
  \frac{\varphi^4 + \xi^4-\psi^4 + 2(\varphi^2-\psi^2)(\psi^2-\xi^2)}{4\varphi^2 \psi^2\xi^2} - \frac{\varphi'\xi'}{\varphi \xi}   &
    \;  \frac{\varphi'(\varphi^2+\psi^2-\xi^2)}{2\varphi^2\psi \xi } + \frac{\xi'(\xi^2+\psi^2-\varphi^2)}{2\varphi \psi\xi^2} -\frac{\psi'}{\varphi \xi} \\[5pt]
    \; \frac{\varphi'(\varphi^2+\psi^2-\xi^2)}{2\varphi^2\psi \xi } + \frac{\xi'(\xi^2+\psi^2-\varphi^2)}{2\varphi \psi\xi^2} -\frac{\psi'}{\varphi \xi}  &
    -\frac{\psi''}{\psi}
 \end{bmatrix},\\[3pt]
  {R}_3 &= \begin{bmatrix}
  \frac{ \varphi^4+\psi^4-\xi^4 + 2(\psi^2-\xi^2)(\xi^2-\varphi^2)}{4\varphi^2 \psi^2\xi^2 } - \frac{\varphi'\psi'}{\varphi\psi}   &
     \frac{\varphi'(\varphi^2+\xi^2-\psi^2)}{2\varphi^2\psi\xi}+ \frac{\psi'(\psi^2+\xi^2-\varphi^2)}{2\varphi\psi^2 \xi}  -\frac{\xi'}{\varphi\psi}  \\[5pt]
      \frac{\varphi'(\varphi^2+\xi^2-\psi^2)}{2\varphi^2\psi\xi}+ \frac{\psi'(\psi^2+\xi^2-\varphi^2)}{2\varphi\psi^2 \xi}  -\frac{\xi'}{\varphi\psi}  &
    -\frac{\xi''}{\xi}
 \end{bmatrix}.\end{aligned}$$

The Hodge star operator $*$ is also clearly block diagonal in the basis ${\mathcal{B}}$, namely, $$\label{eq:defH}
* = \operatorname{diag}(H, H, H), \quad\text{where}\quad H= \begin{bmatrix}
      0 & 1\\
      1 & 0
     \end{bmatrix}.$$ Thus, by the Finsler–Thorpe trick (), such ${R}={\operatorname{diag}}({R}_1,{R}_2,{R}_3)$ as in has $\sec\geq0$, respectively $\sec>0$, if and only if there exists $\tau(r)$ such that ${R}_i+\tau\,H\succeq0$ for $i=1,2,3$, respectively ${R}_i+\tau\,H\succ0$ for $i=1,2,3$.

Diagonal entries in ${R}_i$ are sectional curvatures $\sec(e_i\wedge e_j)=R_{ijij}$ of coordinate planes, while off-diagonal entries are $R_{ijkl}$, with $i,j,k,l$ all distinct, so the Finsler–Thorpe trick states that $\sec\geq0$ and $\sec>0$ are respectively equivalent to the existence of $\tau$ such that all $R_{ijij}\,R_{klkl}- (R_{ijkl}+\tau)^2$ are $\geq0$ and $>0$.

To illustrate the above, note that setting $\varphi,\psi,\xi$ to be the functions in that correspond to the standard metrics in $S^4$ and ${\mathds{C}}P^2$, the blocks $R_i$ become constant: $$\label{eq:can-R-blocks}
\begin{aligned}
   S^4&: \qquad {R}_1={R}_2={R}_3=\begin{bmatrix}
           1 & 0 \\
           0 & 1
          \end{bmatrix},  \\[1pt]
   {\mathds{C}}P^2&: \qquad {R}_1 = {R}_2= \begin{bmatrix}
           1 & -1 \\
           -1 & 1
          \end{bmatrix}, \quad {R}_3 = \begin{bmatrix}
           4 & 2 \\
           2 & 4
          \end{bmatrix}.
\end{aligned}$$ In particular, $\tau$ can be chosen constant, and $R+\tau\,*\succeq0$ if and only if $\tau\in[-1,1]$ for $S^4$, and $\tau\in[0,2]$ for ${\mathds{C}}P^2$, and $R+\tau\,*\succ0$ if and only if $\tau$ is in the open intervals.

Similarly, the curvature of a Grove–Ziller metric with gluing interface $\partial D(B_\pm)$ isometric to $({\mathsf{G}}/{\mathsf{H}},b^2Q|_{\mathfrak{n}})$ and $L=r_{\mathrm{max}}^+ +r_{\mathrm{max}}^-$ can be computed by setting $\varphi,\psi,\xi$ instead to be the functions that make match with , namely (see ) $$\begin{aligned}
 \varphi(r) &= \begin{cases}
 \frac{f(r)\,b \,\sqrt{a}}{\sqrt{f(r)^2 + a \rho^2}}, & \text{ if }  r\in \left(0,r_{\mathrm{max}}^- \right], \text{ where } \rho=\rho_-(b), \; f=f_-, \\  b, & \text{ if } r\in \left[r_{\mathrm{max}}^- , L\right), \end{cases}\nonumber \\[3pt]
 \psi(r)&\equiv b,\label{eqn:f1intermsoff} \\[3pt]
 \xi(r) &= \begin{cases}
b, & \text{ if }  r\in \left(0,r_{\mathrm{max}}^- \right],\\
 \frac{f(L-r)\,b \,\sqrt{a}}{\sqrt{f(L-r)^2 + a \rho^2}},& \text{ if }r \in \left[r_{\mathrm{max}}^- , L\right), \text{ where } \rho=\rho_+(b), \; f=f_+,
 \end{cases}\nonumber
 \end{aligned}$$ as ${\mathfrak{m}}={\mathfrak{n}}_2\oplus {\mathfrak{n}}_3$ and ${\mathfrak{p}}={\mathfrak{n}}_1$ for the disk bundle $D(B_-)$, but $\varphi$ and $\xi$ switch roles on the disk bundle $D(B_+)$, in which ${\mathfrak{m}}={\mathfrak{n}}_1\oplus {\mathfrak{n}}_2$ and ${\mathfrak{p}}= {\mathfrak{n}}_3$. Recall that $f(r)\equiv \tfrac{\sqrt{a}\,\rho}{\sqrt{a-1}}$ for $r_0\leq r\leq r_{\mathrm{max}}$ on each of $D(B_\pm)$, so, in a neighborhood of the gluing interface $r=r_{\mathrm{max}}^-=L-r_{\mathrm{max}}^+$, the functions $\varphi=\psi=\xi$ are all constant and equal to $b$ . In what follows, to simplify the exposition, we shall work with $\varphi,\psi,\xi$ only on the interval $\left(0,r_{\mathrm{max}}^-\right]$, which, at least on $S^4$, determines their values for all $0<r<L$ by setting $r_{\mathrm{max}}^+=r_{\mathrm{max}}^-$ and imposing the additional symmetries , see .

Straightforward computations using imply the following:

\[propn:GZRR\] The curvature operator of the Grove–Ziller metric ; i.e., the metric with $\varphi,\psi,\xi$ as in , for $r\in \left(0,r_{\mathrm{max}}^-\right]$, is $R={\operatorname{diag}}(R_1,R_2,R_3)$, with: $${R}_1 = \begin{bmatrix}
           \frac{4b^2 - 3\varphi^2}{4b^4} & -\frac{\varphi'}{b^2} \\
           -\frac{\varphi'}{b^2} & -\frac{\varphi''}{\varphi}
          \end{bmatrix}, \quad 
  {R}_2 = {R}_3 = \begin{bmatrix}
           \frac{\varphi^2}{4b^4} & \frac{\varphi'}{2b^2} \\
           \frac{\varphi'}{2b^2} & 0
          \end{bmatrix}.$$ In particular, $R+\tau\,*\succeq0$ if and only if $\tau=-\frac{\varphi'}{2b^2}$.

Indeed, it is easy to verify that $\tau=-\frac{\varphi'}{2b^2}$ is the *only* function $\tau(r)$, $r\in \left(0,r_{\mathrm{max}}^-\right]$, such that $R+\tau\,*\succeq0$. Namely, for such $r$, we have that $[{R}_i + \tau H]_{22}\equiv 0$ for both $i=2,3$, and hence $\det({R}_2 + \tau H)=-(\frac{\varphi'}{2b^2}+\tau)^2\geq0$. This pointwise uniqueness of $\tau$ corresponds to the presence of flat planes for the Grove–Ziller metric at every point $\gamma(r)$; e.g., $\sec(e_0\wedge e_2)\equiv 0$ for all $r$. It is interesting to observe how this (forceful) choice of $\tau$ stemming from $R_i+\tau H\succeq0$, $i=2,3$, also satisfies $R_1+\tau H\succeq0$, i.e., how the expression for $\varphi$ in ensures $\det({R}_1 + \tau H) = \big( \frac{4b^2 - 3\varphi^2}{4b^4}\big)\big(-\frac{\varphi''}{\varphi}\big) - \big(\frac{3\varphi'}{2b^2}\big)^2\geq0$.

\[propn:eqnf1\_sec&gt;0\] The function $\varphi(r)$ in the Grove–Ziller metric , given by for $r\in \left(0,r_{\mathrm{max}}^-\right]$, satisfies $(4b^2 - 3\varphi^2)(-\varphi'') - 9\varphi\varphi'^2 \geq 0$ for all $r\in \left(0,r_{\mathrm{max}}^-\right]$.

Solving for $f(r)$ in , we find $f(r)= \frac{\varphi(r)\rho\sqrt{a}}{ \sqrt{ab^2 - \varphi(r)^2}}$; in particular, we have that $\varphi(r)<\sqrt{a}\,b$. Differentiating twice, it follows that: $$\label{eq:f''intermsofphi}
    f'' = \frac{a^{3/2}b^2\rho}{(ab^2 - \varphi^2)^{5/2}} \big( \varphi''(a b^2 - \varphi^2) + 3\varphi\varphi'^2\big).$$ Since $f''\leq 0$, we have $\varphi''(a b^2 - \varphi^2) + 3\varphi\varphi'^2\leq0$, so $(3a b^2 - 3\varphi^2)(-\varphi'') - 9\varphi\varphi'^2 \geq 0$, which implies the desired differential inequality since $a \leq \frac{4}{3}$.

Positively curved metrics near Grove–Ziller metrics {#sec:sec>0gs}
===================================================

In this section, we prove in the Introduction, perturbing arbitrary Grove–Ziller metrics with $\sec\geq0$ on $S^4$ and ${\mathds{C}}P^2$ into cohomogeneity one metrics that we show have $\sec>0$ via the Finsler–Thorpe trick ().

Metric perturbation
-------------------

Let $M^4$ be either $S^4$ or ${\mathds{C}}P^2$, with the cohomogeneity one action of ${\mathsf{G}}={\mathsf{SO}}(3)$ from the previous section. Given a Grove–Ziller metric ${\mathrm g_{\mathrm{GZ}}}$ on $M^4$ with gluing interface isometric to $({\mathsf{G}}/{\mathsf{H}},b^2 Q|_{\mathfrak{n}})$, we have that the length of the circle(s) ${\mathsf{K}}_\pm/{\mathsf{H}}$ endowed with the metric $b^2\,Q|_{{\mathfrak{p}}_\pm}$ is $\rho_\pm(b) = b/|({\mathsf{K}}_\pm)_0\cap {\mathsf{H}}|$, where ${\mathsf{K}}_0$ is the identity component of ${\mathsf{K}}$. From the group diagrams, we compute $|({\mathsf{K}}_\pm)_0\cap {\mathsf{H}}|$ and obtain $\rho_\pm(b) = b/2$ if $M^4=S^4$, while $\rho_-(b) = b$ and $\rho_+(b) = b/2$ if $M^4={\mathds{C}}P^2$. Thus, by , the length $L$ of the orbit space $M/{\mathsf{G}}=[0,L]$ satisfies $L>\frac{\sqrt{a}}{\sqrt{a-1}} \,b$ if $M^4=S^4$, and $L> \frac{3\sqrt{a}}{2\sqrt{a-1}}\, b$ if $M^4={\mathds{C}}P^2$. Rescaling $(M^4,{\mathrm g_{\mathrm{GZ}}})$ so that $L=\frac\pi3$ if $M^4=S^4$, and $L = \frac\pi4$ if $M^4={\mathds{C}}P^2$, we obtain a Grove–Ziller metric ${\mathrm g}_0$ homothetic to ${\mathrm g_{\mathrm{GZ}}}$, with standardized $L$, and whose parameters $a$ and $b$ satisfy $$\label{eq:max-beta}
\textstyle b< \frac\pi3\frac{\sqrt{a-1}}{\sqrt{a}} \;\text{ if }\; M^4=S^4, \quad\text{and}\quad b< \frac\pi6\frac{\sqrt{a-1}}{\sqrt{a}} \;\text{ if }\; M^4={\mathds{C}}P^2.$$ Using , it follows that $r_{\mathrm{max}}^\pm = \frac\pi6$ for $M^4=S^4$, while $r_{\mathrm{max}}^- = \frac{\pi}{6}$ and $r_{\mathrm{max}}^+ = \frac{\pi}{12}$ for $M^4={\mathds{C}}P^2$. Note that $\varphi_1(r)=\xi_1(r)$ precisely at these values of $r=r^-_{\mathrm{max}}$.

Writing ${\mathrm g}_0$ in the form we obtain the functions $\varphi,\psi,\xi$ in , which we decorate with the subindex $_0$, i.e., $\varphi_0,\psi_0,\xi_0$. Similarly, let ${\mathrm g}_1$ be the standard metric on $M^4$, and use a subindex $_1$ to decorate the $\varphi,\psi,\xi$ given in . Now, define: $$\label{eq:def-phis-psis-xis}
\begin{aligned}
 \varphi_s(r) &:= (1-s)\varphi_0(r) + s\,\varphi_1(r),\\
 \psi_s(r) &:=  (1-s)\psi_0(r) + s\,\psi_1(r), \qquad r\in \left[ 0, L \right],\\
  \xi_s(r) &:= (1-s)\,\xi_0(r) + s\,\xi_1(r),
\end{aligned}$$ i.e., linearly interpolate from $\varphi_0,\psi_0,\xi_0$ to $\varphi_1,\psi_1,\xi_1$, and set ${\mathrm g}_s$, $s\in[0,1]$, to be $$\label{eq:gs}
{\mathrm g}_s := {\mathrm{d}}r^2 + \varphi_s(r)^2 \, Q|_{{\mathfrak{n}}_1} + \psi_s(r)^2 \, Q|_{{\mathfrak{n}}_2} + \xi_s(r)^2 \, Q|_{{\mathfrak{n}}_3}, \quad 0<r<L.$$ The functions can be visualized as affine homotopies between .

\[scale=0.75\]

\[ axis x line=middle, axis y line=middle, axis line style = thick, tick style = thick, ymax=2.2, xmax=<span>pi+.1</span>, xtick=<span>pi/2,pi</span>, xticklabels=<span>$r_{\mathrm{max}}^-=\frac\pi6$, $L=\frac\pi3$</span>, ytick=<span>0.8,1</span>, yticklabels=<span>$b$,$\frac{\pi}{6}$</span>, \] ; ; ; ; (3\*pi/4,0.9) node <span>$\color{red}\varphi_0$</span>; (pi/4,0.9) node <span>$\color{black!40!green}\psi_0$</span>; (9\*pi/10,0.45) node <span>$\color{blue}\xi_0$</span>;

\[scale=0.75\]

\[ axis x line=middle, axis y line=middle, axis line style = thick, tick style = thick, ymax=2.2, xmax=<span>3\*pi/4+.1</span>, xtick=<span>pi/2,3\*pi/4</span>, xticklabels=<span>$r_{\mathrm{max}}^-=\frac\pi6$, $L=\frac\pi4$</span>, ytick=<span>0.3,0.5</span>, yticklabels=<span>$b$,$\frac{\pi}{12}$</span>, \] ; ; ; ; (3\*pi/4-.2,0.4) node <span>$\color{red}\varphi_0$</span>; (pi/5+.1,0.4) node <span>$\color{black!40!green}\psi_0$</span>; (3\*pi/4-.1,0.17) node <span>$\color{blue}\xi_0$</span>;

It is a straightforward consequence of that ${\mathrm g}_s$ are smooth metrics:

\[propn:gssmooth\] The ${\mathsf{G}}$-invariant metrics ${\mathrm g}_s$, $s\in [0,1]$, defined on $M^4\setminus (B_-\cup B_+)$ by , extend to smooth metrics on $M^4$, which we also denote by ${\mathrm g}_s$, $s\in [0,1]$.

For simplicity, we focus on the case $M^4=S^4$, and the case $M^4={\mathds{C}}P^2$ is left to the reader. The metrics ${\mathrm g}_s$ are clearly smooth away from the singular orbits, which correspond to $r=0$ and $r=L$. In light of , it suffices to check the smoothness conditions (i)–(iii) in , i.e., those regarding $r=0$.

First, since $\varphi_s^{(k)}(r)=(1-s)\varphi_0^{(k)}(r)+s\,\varphi_1^{(k)}(r)$ for all $k\geq0$, it is clear that $\varphi_s$ satisfies (i), as both $\varphi_0$ and $\varphi_1$ do. Second, if $r\in \left[0,r_{\mathrm{max}}^-\right]$, then $\psi_0(r)=\xi_0(r)=b$, cf. , so $\psi_s(r) = (1-s) b + s\,\psi_1(r)$ and $\xi_s(r) = (1-s)b + s\,\xi_1(r)$, and thus: $$\begin{aligned}
\psi_s(r)^2 + \xi_s(r)^2 &= 2(1-s)^2b^2  + 2s(1-s)b (\psi_1(r) + \xi_1(r)) + s^2 \left( \psi_1(r)^2 + \xi_1(r)^2 \right)\\
  &= 2(1-s)^2b^2 + 2s(1-s)b\,\sqrt{3}\cos r + s^2 \phi_1(r^2) = \widetilde{\phi_1}(r^2),\\
\psi_s(r)^2 - \xi_s(r)^2  &=  2s(1-s)b \, (\psi_1(r) - \xi_1(r)) + s^2 \left( \psi_1(r)^2 - \xi_1(r)^2 \right) \\
  &=  2s(1-s)b\,(-2\sin r) + s^2 \,r\,\phi_2(r^2) = r\,\widetilde{\phi_2}(r^2),
 \end{aligned}$$ where $\widetilde{\phi_k}$, $k=1,2$, are smooth functions, hence (ii) and (iii) are also satisfied.

Let us introduce functions $\Delta_\varphi,\Delta_\psi,\Delta_\xi$ of $r$ so that can be written as $$\label{eqn:Deltai}
\varphi_s=\varphi_0+s\,\Delta_\varphi, \quad \psi_s=\psi_0+s\,\Delta_\psi, \quad \xi_s=\xi_0+s\,\Delta_\xi,$$ i.e., $\Delta_\varphi(r) := \varphi_1(r)-\varphi_0(r)$, and similarly for $\Delta_\psi$ and $\Delta_\xi$. Note that each of these functions is smooth up to $r=0$ and $r=L$; in particular, bounded on $[0,L]$. In the sequel, we take the point of view that $\varphi_s,\psi_s,\xi_s$ are perturbations of $\varphi_0,\psi_0,\xi_0$.

Regularity of perturbation
--------------------------

By , , and , each entry of the curvature operator matrix $R_s$ of ${\mathrm g}_s$ along $\gamma(r)$ is a smooth function $$\label{eq:generic-entry}
 \frac{ P(\varphi_s,\, \psi_s,\,\xi_s,\, \varphi'_s,\, \psi'_s,\, \xi'_s,\, \varphi''_s,\, \psi''_s, \,\xi''_s)}{\varphi_s^2\,\psi_s^2\,\xi_s^2},$$ where $P$ is a polynomial. Note that the ${\mathrm g}_s$-orthonormal basis on which the matrix $R_s$ is being written varies smoothly with $s$. The singularities in at $r=0$ and $r=L$, due to $\varphi_s(0)=0$ and $\xi_s(L)=0$, are removable as a consequence of . This corresponds to the fact that also $P$ vanishes to the appropriate order because $\varphi_s,\psi_s,\xi_s$ satisfy the required smoothness conditions. Moreover, these smoothness conditions imply that equals $$\label{eq:generic-entry2}
 \frac{ P(\varphi_s,\, \psi_s,\,\xi_s,\, \varphi'_s,\, \psi'_s,\, \xi'_s,\, \varphi''_s,\, \psi''_s, \,\xi''_s)}{\varphi_0^2\,\psi_0^2\,\xi_0^2}+Q(s,r)\,s,$$ where $Q$ is continuous. Furthermore, by , the numerator above can be written as a polynomial $\widetilde P$ in the parameter $s$, the functions $\varphi_0,\psi_0,\xi_0$ and their first and second derivatives, and the functions $\Delta_\varphi,\Delta_\psi,\Delta_\xi$ and their first and second derivatives (indicated as $\dots$ below). Thus, and hence are equal to $$\label{eq:generic-entry3}
 \frac{ \widetilde P(s,\, \varphi_0,\, \psi_0,\,\xi_0,\dots, \Delta_\varphi, \, \Delta_\psi,\, \Delta_\xi, \dots)}{\varphi_0^2\,\psi_0^2\,\xi_0^2}+Q(s,r)\,s.$$ In particular, the dependence of the above on $s$ is polynomial in the first term, and smooth on the second. Expanding in $s$, we have $$\widetilde P(s,\, \varphi_0, \psi_0,\xi_0,\dots, \Delta_\varphi, \Delta_\psi, \Delta_\xi, \dots)=\sum_{n=0}^d \widetilde{P}_n(\varphi_0, \psi_0,\xi_0,\dots, \Delta_\varphi, \Delta_\psi,\Delta_\xi, \dots)\,s^n,$$ where $\widetilde{P}_n$ are polynomials. Each coefficient in this sum is a smooth function of $r$ that vanishes at $r=0$ and $r=L$ in such way that the limits of as $r\searrow0$ and $r\nearrow L$ are both finite, so the corresponding coefficients in extend to smooth (hence bounded) functions on $[0,L]$. Thus, $ \widetilde P(s,\, \varphi_0,\, \psi_0,\,\xi_0,\dots, \Delta_\varphi, \, \Delta_\psi,\, \Delta_\xi, \dots)/\varphi_0^2\,\psi_0^2\,\xi_0^2$ can be regarded as a polynomial in the variable $s$ whose coefficients are *continuous* functions of $r$. We will implicitly (and repeatedly) use this fact in what follows.

We use $O(s^n)$, respectively $O(r^m)$, to denote any functions of the form $s^n\, F(s,r)$, respectively $r^m\, F(s,r)$, where $F\colon [0,1]\times[0, L]\to{\mathds{R}}$ is *bounded*.

Positive curvature on the four-sphere
-------------------------------------

To simplify the exposition, we shall focus primarily on the case $M^4=S^4$, in which $r_{\mathrm{max}}^\pm=\frac{L}{2}=\frac\pi6$ and it suffices to verify $\sec>0$ along the geodesic segment $\gamma(r)$ with $r\in \left[0, r_{\mathrm{max}}^- \right]$ due to the additional additional symmetries , cf. . Let $R_s={\operatorname{diag}}\!\big( (R_s)_1,(R_s)_2,(R_s)_3 \big)$ be the curvature operator of $(S^4,{\mathrm g}_s)$ along $\gamma(r)$, given by , where $\varphi,\psi,\xi$ are set to be $\varphi_s,\psi_s,\xi_s$ defined in . As discussed above, $R_s$, $s\in [0,1]$, extends smoothly to $r=0$, and this extension (as well as its entries) will be denoted by the same symbol(s). Clearly, $R_0$ is the curvature operator of the Grove–Ziller metric ${\mathrm g}_0$, so $R_0+\tau_0\,*\succeq0$ for all $r\in \left[0,r_{\mathrm{max}}^-\right]$, where $\tau_0 := -\frac{\varphi_0'}{2b^2}$, see . The proof of hinges on the next:

\[claim-S4\] If $s>0$ is sufficiently small, then $R_s+\tau_s\,*\succ0$ for all $r\in \left[0,r_{\mathrm{max}}^-\right]$, with $$\label{eqn:thorpe_s}
 \tau_s(r) := \tau_0(r)+  \frac{2(\sqrt{3}-b)}{b^3}\,s=  -\frac{\varphi_0'(r)}{2b^2} +  \frac{2(\sqrt{3}-b)}{b^3}\,s.$$

We begin the journey towards observing that certain diagonal entries of $R_s$, which are sectional curvatures with respect to ${\mathrm g}_s$, are positive for all $s\in(0,1]$.

\[propn:sec&gt;0\_1234\] For all $s\in(0,1]$ and $r\in \left[0, r_{\mathrm{max}}^-\right]$, the following hold:

1.  $[(R_s)_i]_{22} = \sec_{{\mathrm g}_s}(e_0\wedge e_i) > 0$ for $1 \leq i \leq 3$;

2.  $[(R_s)_1]_{11} =\sec_{{\mathrm g}_s}(e_2 \wedge e_3)>0$.

As the round metric ${\mathrm g}_1$ has $\sec \equiv1$, we have $\varphi_1''(r) <0$, $\psi_1''(r) <0$, $\xi_1''(r) <0$ by , cf.  and . Thus $\varphi_s''(r) <0$, $\psi_s''(r) <0$, $\xi_s''(r) <0$ for all $s\in(0,1]$ and $r\in\left[0,r_{\mathrm{max}}^-\right]$, which implies, by , that $\sec_{{\mathrm g}_s}(e_0\wedge e_i) > 0$, for $i=2,3$. In the case of $\sec_{{\mathrm g}_s}(e_0\wedge e_1)$, a further argument is required at $r=0$. Namely, using the smoothness conditions, we see that if $s\in (0,1]$, then $$\lim_{r\searrow 0} \sec_{{\mathrm g}_s}(e_0\wedge e_1)(r) = (1-s)\sec_{{\mathrm g}_0}(e_0\wedge e_1)(0) + s\sec_{{\mathrm g}_1}(e_0\wedge e_1)(0) >0,$$ where $(e_0\wedge e_1)(r)$ denotes the $2$-plane in $T_{\gamma(r)}S^4$ spanned by $e_0$ and $e_1$, which concludes the proof of (i). Regarding (ii), if $s\in (0,1]$ and $r \in \left(0, r_{\mathrm{max}}^-\right]$, then $$\varphi_s \leq \xi_s < \psi_s, \quad \xi_s' < 0, \quad \psi_s' \geq 0,$$ which implies that $$\begin{aligned}
\sec_{{\mathrm g}_s}(e_2\wedge e_3) &= \frac{\psi_s^4+\xi_s^4-\varphi_s^4 + 2(\xi_s^2-\varphi_s^2)(\varphi_s^2-\psi_s^2)}{4\,\varphi_s^2\, \psi_s^2\,\xi_s^2 } - \frac{\psi_s'\xi_s'}{\psi_s\xi_s} \\
&= \frac{(\xi_s^2 - \psi_s^2)^2}{4\,\varphi_s^2\,\psi_s^2 \,\xi_s^2} +\frac{2\psi_s^2 - \varphi_s^2}{4\,\psi_s^2 \,\xi_s^2} +\frac{\xi_s^2-\varphi_s^2}{2\,\psi_s^2 \,\xi_s^2} - \frac{\psi_s'\xi_s'}{\psi_s\xi_s} \geq\frac{b^2}{4\psi_s^2\,\xi_s^2},
  \end{aligned}$$ since $2\psi_s^2-\varphi_s^2\geq \psi_s^2$ and $\psi_s\geq \psi_0\equiv b$ is uniformly bounded from below.

Let us introduce functions $\eta_i,\mu_i,\nu_i$, $i=1,2,3$, such that the blocks of the curvature operator ${R}_s={\operatorname{diag}}\!\big(({R}_s)_1,({R}_s)_2,({R}_s)_3\big)$ of ${\mathrm g}_s$ can be written as a perturbation $$\label{eq:def-etai-mui-nui}
 \phantom{, \qquad i = 1,2,3,}
  ({R}_s)_i = ({R}_0)_i +  \begin{bmatrix}
    \eta_i(s,r) & \mu_i(s,r)\\
    \mu_i(s,r) & \nu_i(s,r)
  \end{bmatrix}, \qquad i = 1,2,3,$$ of the blocks of the curvature operator ${R}_0={\operatorname{diag}}\!\big(({R}_0)_1,({R}_0)_2,({R}_0)_3\big)$ of the Grove–Ziller metric ${\mathrm g}_0$. Recall that, for $r\in\left(0,r_{\mathrm{max}}^-\right]$, these blocks $({R}_0)_i$ are computed in , setting $\varphi=\varphi_0$, i.e., $\varphi$ is given by . Clearly, each of $\eta_i,\mu_i,\nu_i$ is $O(s^n)$ for some $n\geq1$.

### First block

We first analyze the block $i=1$ of the matrices $R_s$ and $R_s+\tau_s\,*$.

\[propn:Rs1-new\] For all $r\in \left[0, r_{\mathrm{max}}^-\right]$, the entries of $({R}_s)_1$ satisfy: $$\begin{aligned}
   \eta_1(s, r) &=  \left(  \frac{3\varphi_0}{2b^5} (\varphi_0 (\Delta_\psi + \Delta_\xi)-b\Delta_\varphi ) - \frac{\Delta_\psi + \Delta_\xi}{b^3} \right)s + O(s^2),\\
   \mu_1(s,r) &= \left( \frac{\varphi_0(\psi_1' + \xi_1')}{2b^3} - \frac{\Delta_\varphi'}{b^2} + \frac{\varphi_0'}{b^3}(\Delta_\psi + \Delta_\xi) \right)s + O(s^2),\\
   \nu_1(s,r) &=  \left( \frac{-\varphi_1''\varphi_0 + \varphi_0''\varphi_1}{\varphi_0^2} \right)s + O(s^2).
  \end{aligned}$$

First, let us consider $\eta_1$. From , $$\begin{aligned}
 [({R}_s)_1]_{11} &= \frac{\psi_s^4 +\xi_s^4 - \varphi_s^4 + 2(\xi_s^2 - \varphi_s^2)(\varphi_s^2 - \psi_s^2)}{4\varphi_s^2\,\psi_s^2\,\xi_s^2} - \frac{\psi_s'\xi_s'}{\psi_s\xi_s}\\
 &= \frac{(\xi_s^2 - \psi_s^2)^2}{4\varphi_s^2\,\psi_s^2\,\xi_s^2} - \frac{3\varphi_s^2}{4\psi_s^2\,\xi_s^2} + \frac{\xi_s^2 + \psi_s^2}{2\psi_s^2\,\xi_s^2} - \frac{\psi_s'\xi_s'}{\psi_s\xi_s}.\end{aligned}$$ We analyze these four terms separately using , as follows $$\begin{aligned}
- \frac{3\varphi_s^2}{4\psi_s^2\,\xi_s^2} &= -\frac{3\varphi_0^2}{4b^4} - \frac{3\varphi_0}{2b^5} (b\Delta_\varphi - \varphi_0(\Delta_\psi +\Delta_\xi) )s + O(s^2),\\
 \frac{\xi_s^2 + \psi_s^2}{2\psi_s^2\,\xi_s^2}  &= \frac{1}{b^2} - \frac{\Delta_\psi + \Delta_\xi}{b^3}\,s + O(s^2),\quad  \frac{(\xi_s^2 - \psi_s^2)^2}{4\varphi_s^2\,\psi_s^2\,\xi_s^2} =  O(s^2), \quad - \frac{\psi_s'\xi_s'}{\psi_s\xi_s} = O(s^2).\end{aligned}$$ Therefore, adding the above together, we find: $$[({R}_s)_1]_{11} = \frac{4b^2 - 3\varphi_0^2}{4b^4} +\left(  \frac{3\varphi_0}{2b^2} (\varphi_0 (\Delta_\psi + \Delta_\xi)-b \Delta_\varphi ) - \frac{\Delta_\psi + \Delta_\xi}{b^3} \right)s + O(s^2),$$ which establishes the claimed expansion of $\eta_1(s,r)= [({R}_s)_1]_{11} - \frac{4b^2 - 3\varphi_0^2}{4b^4}$, cf. .

Next, consider $\mu_1$. From , $$\begin{aligned}
 [({R}_s)_1]_{12} &= \frac{\xi_s'(\xi_s^2+\varphi_s^2-\psi_s^2)}{2\varphi_s\,\psi_s\,\xi_s^2} + \frac{\psi_s'(\varphi_s^2+\psi_s^2-\xi_s^2)}{2\varphi_s\,\psi_s^2\, \xi_s} - \frac{\varphi_s'}{\psi_s\,\xi_s}\\ 
 &= \frac{(\xi_s^2 - \psi_s^2)(\xi_s'\psi_s - \psi_s'\xi_s)}{2\varphi_s\,\psi_s^2\,\xi_s^2} + \frac{\varphi_s(\xi_s'\psi_s + \psi_s'\xi_s)}{2\psi_s^2\,\xi_s^2} - \frac{\varphi_s'}{\psi_s\,\xi_s}.\end{aligned}$$ We analyze these three terms separately, using , as before: $$\begin{gathered}
 \frac{(\xi_s^2 - \psi_s^2)(\xi_s'\psi_s - \psi_s'\xi_s)}{2\varphi_s\,\psi_s^2\,\xi_s^2} = O(s^2), \quad
\frac{\varphi_s(\xi_s'\psi_s + \psi_s'\xi_s)}{2\psi_s^2\,\xi_s^2} 
 = \frac{\varphi_0(\psi_1' + \xi_1')}{2b^3}\,s + O(s^2),\\
-\frac{\varphi_s'}{\psi_s\,\xi_s} = -\frac{\varphi_0'}{b^2} + \left( \frac{\varphi_0'(\Delta_\psi + \Delta_\xi)}{b^3}-\frac{\Delta_\varphi'}{b^2} \right)s + O(s^2).\end{gathered}$$ Thus, adding the above, we have: $$[({R}_s)_1]_{12} = -\frac{\varphi_0'}{b^2}  + \left( \frac{\varphi_0(\psi_1' + \xi_1')}{2b^3} - \frac{\Delta_\varphi'}{b^2} + \frac{\varphi_0'(\Delta_\psi + \Delta_\xi)}{b^3} \right)s + O(s^2),$$ which establishes the claimed expansion of $\mu_1(s,r)=[({R}_s)_1]_{12} +\frac{\varphi_0'}{b^2}$, cf. .

Finally, let us consider $\nu_1$. From , we have: $$[({R}_s)_1]_{22} = -\frac{\varphi_s''}{\varphi_s}
 = -\frac{\varphi_0''}{\varphi_0} + \left( \frac{-\varphi_1''\varphi_0 + \varphi_0''\varphi_1}{\varphi_0^2} \right)s + O(s^2),$$ which establishes the claimed expansion of $\nu_1(s,r)= [({R}_s)_1]_{22}+\frac{\varphi_0''}{\varphi_0}$, cf. .

\[prop:Rs1-positive\] If $s>0$ is sufficiently small, then the matrix $$  ({R}_s)_1 + \tau_s H = \begin{bmatrix}
    \frac{4b^2 - 3\varphi_0^2}{4b^4} +\eta_1(s,r) & -\frac{3\varphi_0'}{2b^2}+\mu_1(s,r)+\frac{2(\sqrt{3}-b)}{b^3}s \\
  -\frac{3\varphi_0'}{2b^2}+\mu_1(s,r)+\frac{2(\sqrt{3}-b)}{b^3}s & -\frac{\varphi_0''}{\varphi_0}+\nu_1(s,r)
  \end{bmatrix}$$ is positive-definite for all $r\in\left[0,r_{\mathrm{max}}^-\right]$.

The expression above for $({R}_s)_1 + \tau_s H$ follows from , as well as , , and . From (ii), we know that $[({R}_s)_1]_{11} > 0$ for all $s\in(0,1]$ and $r\in\left[0,r_{\mathrm{max}}^-\right]$. So, by Sylvester's criterion, it suffices to show that if $s>0$ is sufficiently small, then the following is positive: $$\begin{aligned}
   \det\!\big(({R}_s)_1 + \tau_s H\big) &=   \left(\frac{4b^2 - 3\varphi_0^2}{4b^4}\right) \left( -\frac{\varphi_0''}{\varphi_0} \right) - \left( \frac{3\varphi_0'}{2b^2} \right)^2  -\frac{\varphi_0''}{\varphi_0}  \,\eta_1(s,r)\\
   &  \quad +   \frac{4b^2 - 3\varphi_0^2}{4b^4} \, \nu_1(s,r) + \frac{3\varphi_0'}{b^2} \left(\mu_1(s,r)+\frac{2(\sqrt{3}-b)}{b^3}s\right)\\
  & \quad + \eta_1(s,r)\, \nu_1(s,r) - \left(\mu_1(s,r)+\frac{2(\sqrt{3}-b)}{b^3}s\right)^2.
 \end{aligned}$$ By , we have $\det\!\big(({R}_s)_1 + \tau_s H\big)= A(r) + B(r) \, s+ O(s^2)$, where $$\begin{aligned}
 A(r) &:=  \left(\frac{4b^2 - 3\varphi_0^2}{4b^4}\right) \left( -\frac{\varphi_0''}{\varphi_0} \right) - \left( \frac{3\varphi_0'}{2b^2} \right)^2,\\
B(r) &:=  \left( -\frac{\varphi_0''}{\varphi_0} \right) \left( \frac{3\varphi_0}{2b^5} ( \varphi_0(\Delta_\psi +  \Delta_\xi) -b \Delta_\varphi) - \frac{\Delta_\psi + \Delta_\xi}{b^3}  \right)\\
 &   \quad + \left( \frac{4b^2 - 3\varphi_0^2}{4b^4} \right) \left( \frac{-\varphi_1''\varphi_0 + \varphi_0''\varphi_1}{\varphi_0^2} \right) \\
  & \quad + \frac{3\varphi_0'}{b^2} \, \left( \frac{\varphi_0( \psi_1'+\xi_1')}{2b^3} - \frac{\Delta_\varphi'}{b^2} + \frac{\varphi_0'}{b^3}(\Delta_\psi + \Delta_\xi)  + \frac{2(\sqrt{3}-b)}{b^3}\right).\end{aligned}$$

Note that $A(r)\geq0$ if $r\in\left[0,r_{\mathrm{max}}^-\right]$ by , but $A(r) \equiv 0$ near $r = r_{\mathrm{max}}^-$. We claim that there exist $0<r_*<r_{\mathrm{max}}^-$ and constants $\alpha>0$ and $\beta>0$ such that $$\label{eq:claim-a-b}
\begin{aligned}
&A(r)\geq \alpha>0 \mbox{ for all } 0\leq r\leq r_*, \\
&B(r)\geq \beta>0 \mbox{ for all } r_*\leq r\leq r_{\mathrm{max}}^-,
\end{aligned}$$ from which it clearly follows that $\det\!\big(({R}_s)_1 + \tau_s H\big) >0$ for all $r\in\left[0,r_{\mathrm{max}}^-\right]$ and sufficiently small $s>0$, as desired. Recall that there exists $0< r_0 < r_{\mathrm{max}}^-$ so that:

-   for all $r\in (0,r_0)$, we have $\varphi_0'(r) > 0$ and $\varphi_0''(r)<0$,

-   for all $r\in \left[r_0,r_{\mathrm{max}}^-\right]$, we have $\varphi_0(r) = b$, and hence $\varphi_0'(r) = \varphi_0''(r) =0$,

cf.  and the Grove–Ziller construction (). Moreover, for all $\varepsilon >0$, there exists $0<r_*<r_0$, such that for $r \in \left[r_*, r_{\mathrm{max}}^-\right]$, we have: $$\label{eqn:f1epsilon}
  0 \leq \varphi_0'(r) <\varepsilon, \quad 0 \leq -\varphi_0''(r) < \varepsilon, \;\;\mbox{and}\;\; b - \varepsilon < \varphi_0(r) \leq b, $$ and these inequalities are strict on $[r_*, r_0)$. Thus, choosing $\varepsilon >0$ sufficiently small, we have that for all $r\in \left[r_*, r_{\mathrm{max}}^-\right]$, $$\frac{-\varphi_1''\varphi_0 + \varphi_0''\varphi_1}{\varphi_0^2}
      = \frac{(2\sin r)(\varphi_0 + \varphi_0'')}{\varphi_0^2}
  \geq  \frac{(2\sin r)(b - 2\varepsilon)}{b^2} >  \frac{1}{4b}.$$ Furthermore, by continuity, the following are uniformly bounded on $r\in \left[r_*,r_{\mathrm{max}}^-\right]$, $$\begin{aligned}
  \left| -\frac{1}{\varphi_0}  \left(  \frac{3\varphi_0 }{2b^5}(\varphi_0(\Delta_\psi +\Delta_\xi)-b \Delta_\varphi ) - \frac{\Delta_\psi + \Delta_\xi}{b^3} \right) \right| < C_1,\\
  \left| \frac{3}{b^2}  \left( \frac{\varphi_0(\psi_1'+\xi_1')}{2b^3} -\frac{\Delta_\varphi'}{b^2} + \frac{\varphi_0'(\Delta_\psi + \Delta_\xi)}{b^3} + \frac{2(\sqrt{3}-b)}{b^3} \right) \right| < C_2,
 \end{aligned}$$ where $C_1$ and $C_2$ are constants independent of $r_*$; and $\left( \frac{4b^2 - 3\varphi_0^2}{4b^4} \right) \geq  \frac{1}{4b^2}$ by . Putting the above together, and making $\varepsilon>0$ even smaller if needed, we conclude $$B(r) > -\varepsilon \, C_1 + \tfrac{1}{16b^3} - \varepsilon \, C_2 = \tfrac{1}{16b^3} - \varepsilon\,(C_1 + C_2) > \beta > 0$$ for all $r \in \left[r_*, r_{\mathrm{max}}^- \right]$, where, e.g., $\beta=\tfrac{1}{32b^3}$. Finally, in order to prove the inequality regarding $A(r)$ in , recall there exists $c >0$ such that $\sec_{{\mathrm g}_{D^2}} \geq c>0$ for all $r\in [0, r_*]$, by . From , in the proof of , we have that $$\sec_{{\mathrm g}_{D^2}} = -\frac{f''}{f}  = \frac{ab^2}{(ab^2 - \varphi_0^2)^2}\, \frac{(-\varphi_0'')(ab^2 - \varphi_0^2) - 3\varphi_0\varphi_0'^2}{\varphi_0},$$ from which it follows that $$\frac{3(ab^2 - \varphi_0^2)^2}{ab^2}\,\sec_{{\mathrm g}_{D^2}} =
  3\left( -\frac{\varphi_0''}{\varphi_0} \right) (ab^2 - \varphi_0^2) - 9\varphi_0'^2 \leq \left( -\frac{\varphi_0''}{\varphi_0} \right)(4b^2 - 3\varphi_0^2) - 9\varphi_0'^2,$$ because $1<a\leq\frac43$. Therefore, as $\varphi_0(r)<\sqrt{a}\,b$ for all $r$, there exists $\alpha>0$ so that $$A(r) \geq \frac{3}{4} \, \frac{(ab^2 - \varphi_0^2)^2}{ab^2}\, \sec_{{\mathrm g}_{D^2}} \geq \frac{3}{4} \, \frac{(a b^2 - \varphi_0^2)^2}{ab^2}\, c > \alpha > 0, \;\; \mbox{ for all } r\in [0, r_*].\qedhere$$

### Remaining blocks

We now handle the remaining blocks $i=2,3$.

\[propn:Rs23-new\] For all $r\in \left[0, r_{\mathrm{max}}^-\right]$, the entries of $({R}_s)_i$, for $i=2,3$, satisfy: $$\begin{aligned}
   \eta_i(s, r) &= \left( \tfrac{\sqrt{3}}{b} + O(r) \right)s + O(s^2),
   &\mu_i(s,r) &= \left(- \tfrac{2(\sqrt{3}-b)}{b^3} + O(r) \right)s + O(s^2),\\
  & &\nu_i(s,r)&= \left( \tfrac{\sqrt{3}}{b} + O(r) \right)s + O(s^2).
  \end{aligned}$$

First, let us consider $\eta_2$. From , $$[{({R}_s)_2}]_{11} = \frac{\varphi_s^2}{4 \psi_s^2\,\xi_s^2} + \frac{\psi_s^2 - \xi_s^2}{2 \psi_s^2\,\xi_s^2} + \frac{\xi_s^4 + 2\xi_s^2 \psi_s^2 - 3\psi_s^4 - 4\varphi_s\psi_s^2\xi_s\varphi_s'\xi_s'}{4\varphi_s^2 \,\psi_s^2\,\xi_s^2}.$$ We analyze these three terms separately using . The first two satisfy $$\frac{\varphi_s^2}{4 \psi_s^2\,\xi_s^2 } = \frac{\varphi_0^2}{4b^4} + s\, O(r^2) + O(s^2), \quad \text{ and }\quad   \frac{\psi_s^2 - \xi_s^2}{2 \psi_s^2\,\xi_s^2 }= s\, O(r) + O(s^2),$$ while the third satisfies $$\begin{aligned}
  \frac{\xi_s^4 + 2 \psi_s^2\xi_s^2 - 3\psi_s^4 - 4\varphi_s\psi_s^2\xi_s\varphi_s'\xi_s'}{4\varphi_s^2\,  \psi_s^2\,\xi_s^2} &=\frac{2(\Delta_\xi - \Delta_\psi) - \varphi_0\varphi_0'\Delta_\xi'}{b \varphi_0^2} \, s + O(s^2)\\
  &= \left(\tfrac{\sqrt{3}}{b} + O(r)\right)\,s + O(s^2),
 \end{aligned}$$ since $\displaystyle\lim_{r\searrow 0} \tfrac{2(\Delta_\xi - \Delta_\psi) - \varphi_0\varphi_0'\Delta_\xi'}{\varphi_0^2} = \sqrt{3}$, by L'Hôpital's rule and (i).

Altogether, the above yields $ [({R}_s)_2]_{11} = \frac{\varphi_0^2}{4b^4} + \left( \frac{\sqrt{3}}{b} + O(r) \right)s + O(s^2)$, and hence establishes the claimed expansion of $\eta_2(s,r) = [({R}_s)_2]_{11} -\tfrac{\varphi_0^2}{4b^4}$, cf. .

Second, the proof that $\eta_3$ has the same expansion as $\eta_2$ is similar. Namely, $$\begin{aligned}
  [({R}_s)_3]_{11} &= \frac{\varphi_s^2}{4\psi_s^2\,\xi_s^2 } + \frac{(\psi_s^2 - \xi_s^2)(\psi_s^2 + 3\xi_s^2 - 2\varphi_s^2) - 4\varphi_s\psi_s\xi_s^2\varphi_s'\psi_s'}{4\varphi_s^2\, \psi_s^2\, \xi_s^2},
 \end{aligned}$$ where the first term was already considered above, and the second term satisfies $$\frac{(\psi_s^2 - \xi_s^2)(\psi_s^2 + 3\xi_s^2 - 2\varphi_s^2) - 4\varphi_s\psi_s\xi_s^2\varphi_s'\psi_s'}{4\varphi_s^2\,\psi_s^2 \,\xi_s^2 }\\
    =\left( \frac{\sqrt{3}}{b} + O(r) \right)s + O(s^2),$$ by similar considerations involving L'Hôpital's rule and (i). Thus, $\eta_3(s,r) = [({R}_s)_3]_{11}  - \tfrac{\varphi_0^2}{4b^4} = \left( \frac{\sqrt{3}}{b} + O(r) \right)s + O(s^2)$, cf. .

Next, consider $\mu_2$. From , $$[({R}_s)_2]_{12}  = \frac{\varphi_s'}{2\psi_s\,\xi_s} + \frac{\varphi_s'\xi_s(\psi_s^2 - \xi_s^2) + \varphi_s\xi_s'(\xi_s^2+\psi_s^2-\varphi_s^2) - 2\varphi_s\psi_s\xi_s\psi_s'}{2\varphi_s^2\,\psi_s\,\xi_s^2}$$ The first term above satisfies $$\frac{\varphi_s'}{2\xi_s\psi_s} = \frac{\varphi_0'}{2b^2} +\left( O(r^2) - \frac{2(\sqrt{3}-b)}{b^3} \right)s + O(s^2),$$ while the second satisfies $$\frac{\varphi_s'\xi_s(\psi_s^2 - \xi_s^2) + \varphi_s\xi_s'(\xi_s^2+\psi_s^2-\varphi_s^2) - 2\varphi_s\psi_s\xi_s\psi_s'}{2\varphi_s^2\,\psi_s\,\xi_s^2}
  = s\, O(r) + O(s^2).$$ So, $\mu_2(s,r) = [({R}_s)_2]_{12} -\tfrac{\varphi_0'}{2b^2}=\left(- \frac{2(\sqrt{3}-b)}{b^3} + O(r) \right)s + O(s^2)$, cf. . The proof that $\mu_3$ has the same expansion as $\mu_2$ is analogous, and left to the reader.

Finally, let us consider $\nu_2$ and $\nu_3$. From and , we have $$\nu_2(s,r) =[({R}_s)_2]_{22} =-\tfrac{\psi_s''}{\psi_s}\quad\text{ and } \quad \nu_3(s,r) =[({R}_s)_3]_{22} = -\tfrac{\xi_s''}{\xi_s}.$$ By , we have $\psi_s''  = \Delta_\psi''  \, s= \psi_1'' \,s$ and $\xi_s''  = \Delta_\xi'' \,s  = \xi_1'' \,s$, so $$\nu_2(s,r) = \left( \tfrac{\sqrt{3}}{b} + O(r) \right)s + O(s^2), \; \text{ and }\; \nu_3(s,r) = \left( \tfrac{\sqrt{3}}{b} + O(r) \right)s + O(s^2).\qedhere$$

\[prop:Rs23-positive\] If $s>0$ is sufficiently small, then the matrices $$\label{eq:modified-blocks-23}
  ({R}_s)_i + \tau_s H = \begin{bmatrix}
    \frac{\varphi_0^2}{4b^4} + \eta_i(s,r) & \mu_i(s,r)+\frac{2(\sqrt3-b)}{b^3}s\\
    \mu_i(s,r)+\frac{2(\sqrt3-b)}{b^3}s & \nu_i(s,r)
  \end{bmatrix}, \quad i=2,3,$$ are positive-definite for all $r\in\left[0, r_{\mathrm{max}}^-\right]$.

The expression for $({R}_s)_i + \tau_s H$, $i=2,3$, follows from , as well as , , and . First, consider the $(1,1)$-entry of these matrices: $$\phantom{\quad \text{ for }i=2,3.}[({R}_s)_i]_{11} = \tfrac{\varphi_0^2}{4b^4} + \left( \tfrac{\sqrt{3}}{b} + O(r) \right)s + O(s^2), \quad \text{ for }i=2,3,$$ cf. . Since $\varphi_0(r)>0$ away from $r=0$, and the $O(s)$ part of the above is uniformly positive near $r=0$, it follows that $[({R}_s)_i]_{11}>0$ for all $r\in\left[0, r_{\mathrm{max}}^-\right]$ and $i=2,3$, provided $s>0$ is sufficiently small.

Second, let us analyze the determinant of . By , $$\begin{aligned}
  \eta_i(s,r)  \nu_i(s,r) &= \left( \tfrac{3}{b^2} + O(r) \right)s^2 + O(s^3),\\
   \mu_i(s,r)+\tfrac{2(\sqrt3-b)}{b^3}s &=   s\, O(r) + O(s^2).
 \end{aligned}$$ Thus, using that $\nu_i(s,r)=[(R_s)_i]_{22}$, for $i=2,3$, we have: $$\begin{aligned}
  \det\!\big( ({R}_s)_i + \tau_s H\big) &= \nu_i(s,r)\, \tfrac{\varphi_0^2}{4b^4} + \left( \tfrac{3}{b^2} + O(r) \right)s^2  + O(s^3)\\
  &=  [(R_s)_i]_{22} \, \tfrac{\varphi_0^2}{4b^4} + \left( \tfrac{3}{b^2} + O(r) \right)s^2 + O(s^3).
 \end{aligned}$$ By (i), the $O(s)$ part of the above is positive for $r\in\left(0,r_{\mathrm{max}}^- \right]$, but vanishes at $r= 0$, as $\varphi_0(0)=0$. Since the $O(s^2)$ part has a positive limit as $r\searrow 0$, we have that $\det\!\big( ({R}_s)_i+ \tau_s H\big) >0$ for all $r\in \left[0, r_{\mathrm{max}}^-\right]$ and $i=2,3$, if $s>0$ is sufficiently small. Positive-definiteness now follows from Sylvester's criterion.

The above imply , since $R_s+\tau_s\,*$ is block diagonal with blocks $(R_s)_i+\tau_s H$, $i=1,2,3$, see and . In turn, and the Finsler–Thorpe trick () imply that $\sec_{{\mathrm g}_s}>0$ for sufficiently small $s>0$. This proves for $M^4=S^4$; since, if the original Grove–Ziller metric ${\mathrm g_{\mathrm{GZ}}}$ was rescaled as ${\mathrm g}_0=\lambda^2\,{\mathrm g_{\mathrm{GZ}}}$ to standardize $L=\frac\pi3$, then $\lambda^{-2}\,{\mathrm g}_s$ has $\sec>0$ and is arbitrarily $C^\infty$-close to ${\mathrm g_{\mathrm{GZ}}}$ for $s>0$ sufficiently small.

Positive curvature on the complex projective plane
--------------------------------------------------

We now briefly discuss the proof of for $M^4={\mathds{C}}P^2$. Recall that, in this case, $L=\tfrac{\pi}{4}$, with $r_{\mathrm{max}}^-=\frac\pi6$ and $r_{\mathrm{max}}^+=\frac{\pi}{12}$. Differently from $S^4$, for $M^4={\mathds{C}}P^2$, the situation on the intervals $[0,r_{\mathrm{max}}^-]=\left[0, \tfrac{\pi}{6}\right]$ and $[r_{\mathrm{max}}^-,L]=\left[\tfrac{\pi}{6},\tfrac{\pi}{4}\right]$ has to be analyzed separately, cf. .

Denoting by $R_0$ the curvature operator of the Grove–Ziller metric ${\mathrm g}_0$ on ${\mathds{C}}P^2$, the function $\tau_0\colon [0,L]\to{\mathds{R}}$ so that $R_0+\tau_0\,*\succeq0$ for all $r\in [0,L]$ is given by $$\tau_0(r) = \begin{cases}
      -\frac{\varphi_0'(r)}{2b^2}, & \text{ if } r \in \left[0,r_{\mathrm{max}}^-\right], \\[5pt]
      -\frac{\xi_0'(r)}{2b^2}, & \text{ if } r \in \left[r_{\mathrm{max}}^-,L\right],
  \end{cases}$$ cf. . Note that $\varphi_0'=\xi_0'=0$ near $r=r_{\mathrm{max}}^-$. The proof of follows in the same way as in the case $M^4=S^4$ above, replacing with:

\[claim-CP2\] If $s>0$ is sufficiently small, then $R_s+\tau_s\,*\succ0$ for all $r\in \left[0,L\right]$, where $$\tau_s(r) := 
   \begin{cases}
      -\frac{\varphi_0'(r)}{2b^2} + \left(\frac{3}{2b} +\frac{1-b}{b^3}\right)s, & \text{ if } r \in \left[0,r_{\mathrm{max}}^-\right], \\[5pt]
            -\frac{\xi_0'(r)}{2b^2} +  \tfrac{\sqrt{2}\, -\, 2b}{b^3} \, s, & \text{ if } r \in  \left(r_{\mathrm{max}}^-,L\right].
  \end{cases}$$

Similarly to in , the above function $\tau_s$ is obtained from $\tau_0$ by adding a locally constant multiple of $s$. This $O(s)$ perturbation is not constant as in the case of $M^4=S^4$, and, as a result, $\tau_s(r)$ is *discontinuous* at $r=r_{\mathrm{max}}^-$ for all $s>0$. Nevertheless, the application of the Finsler–Thorpe trick () is pointwise and no regularity is needed. A posteriori, a continuous function $\widetilde\tau_s(r)$ such that $R_s+\widetilde{\tau_s}\,*\succ0$ for all sufficiently small $s>0$ can be chosen, e.g., as the midpoint $\widetilde\tau_s(r)=\frac{1}{2}(\tau_{\mathrm{min}} +\tau_{\mathrm{max}})$ of $[\tau_{\mathrm{min}},\tau_{\mathrm{max}}]$ for each $r\in [0,L]$, see .

The proof of follows the same template from , relying on expansions in $s$ of the functions $\eta_i,\mu_i,\nu_i$, cf. . The statement of , regarding $i=1$ and $r\in [0, r_{\mathrm{max}}^-]$, holds *tout court* for ${\mathds{C}}P^2$, since the smoothness conditions of $\varphi,\psi,\xi$ at $r=0$ are not used in the proof. The case of $i=3$ and $r\in [r_{\mathrm{max}}^-,L]$ is analogous. The replacement for is the following:

\[propn:RsCP2\] For $r\in \left[0, r_{\mathrm{max}}^-\right]$, the entries of $(R_s)_i$, $i=2,3$, satisfy: $$\begin{aligned}
  \eta_2(s,r) &= \left(\tfrac{1}{b} + O(r)\right)s + O(s^2), &  \mu_2(s,r) &= \left(-\tfrac{3}{2b} -\tfrac{1-b}{b^3} + O(r)\right)s + O(s^2),\\
  & &\nu_3(s,r) &= \left(\tfrac{4}{b} + O(r)\right)s + O(s^2),\\
  \eta_3(s,r) &= \left(\tfrac{4}{b} + O(r)\right)s + O(s^2), & \mu_3(s,r) &= \left( \tfrac{3}{2b} -\tfrac{1-b}{b^3}+ O(r)\right)s + O(s^2),\\
  &  &\nu_2(s,r) &=\left(\tfrac{1}{b} + O(r)\right)s  + O(s^2).
 \end{aligned}$$ For $r\in \left[r_{\mathrm{max}}^-,L\right]$, setting $z=L-r$, the entries of $(R_s)_i$, $i=1,2$, satisfy: $$\begin{aligned}
\eta_i(s,z) &= \left(\tfrac{1}{b\sqrt{2}} + O(z)\right)s + O(s^2), &  \mu_i(s,z) &= \left( -\tfrac{\sqrt{2} - 2b}{b^3} + O(z)\right)s + O(s^2),  \\
 & &\nu_i(s,z) &= \left(\tfrac{1}{b\sqrt{2}} + O(z)\right)s + O(s^2).
 \end{aligned}$$

The proof of is totally analogous to that of ; noting that, in terms of $z = L - r\in\left[0,r_{\mathrm{max}}^+\right]$, the functions $\varphi_1,\psi_1,\xi_1$ are: $$\textstyle \varphi_1(z) = \frac{1}{\sqrt{2}}\left(\cos z - \sin z\right), \quad \psi_1(z) = \frac{1}{\sqrt{2}}\left(\cos z + \sin z\right), \quad \xi_1(z) = \sin 2z.$$ Finally, similarly to , it can be shown that $(R_s)_i+\tau_sH$, $i=1,2,3$, are positive-definite for all $r\in [0,L]$ and $s>0$ sufficiently small, which proves (and hence ) for ${\mathds{C}}P^2$. Details are left to the reader.

Positive turns negative {#sec:pos-neg}
=======================

In this section, we prove , using the fact that Grove–Ziller metrics on $S^4$ and ${\mathds{C}}P^2$ immediately acquire negatively curved planes under Ricci flow @bettiol-krishnan1, together with , and continuous dependence on initial data @BGI20.

\[Proof of \] Let $M^4$ be either $S^4$ or ${\mathds{C}}P^2$, and consider the $1$-parameter family of metrics ${\mathrm g}_s$ on $M^4$, defined in , such that ${\mathrm g}_0$ is a Grove–Ziller metric and ${\mathrm g}_1$ is either the round metric or the Fubini–Study metric, accordingly. From , the metrics ${\mathrm g}_s$ are smooth, and it is evident from and that, for all $k\geq0$ and $0<\alpha<1$, there exists a constant $\lambda_{k,\alpha}>0$ such that $$\label{eq:gs-g0}
    \phantom{\quad \text{ for all } 0\leq s\leq 1.}\|{\mathrm g}_s-{\mathrm g}_0\|_{C^{k,\alpha}}\leq \lambda_{k,\alpha}\, s, \quad \text{ for all } 0\leq s\leq 1,$$ where $\|\cdot\|_{C^{k,\alpha}}$ denotes the Hölder norm on sections of the bundle $E={\operatorname{Sym}}^2 TM^4$ with respect to a fixed background metric. For $0\leq s\leq 1$, let ${\mathrm g}_s(t)$, $0\leq t < T({\mathrm g}_s)$, be the maximal solution to Ricci flow starting at ${\mathrm g}_s(0)={\mathrm g}_s$, where $0<T({\mathrm g}_s)\leq +\infty$ denotes the maximal (smooth) existence time of the flow. For all $0\leq s\leq 1$ and $0\leq t<T({\mathrm g}_s)$, we have that ${\mathrm g}_s(t)\in C^\infty(E)$, so ${\mathrm g}_s(t)$ is in the proper closed subspace $h^{k,\alpha}(E) \subset C^{k,\alpha}(E)$ for all $k\geq0$ and $0<\alpha<1$, in the notation of @BGI20. From the main theorem in @bettiol-krishnan1, there exist a $2$-plane $\sigma$ tangent to $M^4$ and $t_0>0$ such that $\sec_{{\mathrm g}_0}(\sigma)=0$ and $\sec_{{\mathrm g}_0(t)}(\sigma)<0$ for all $0<t<t_0$. Fix $0<t_*<t_0$, and let $\delta>0$ be such that $\sec_{\mathrm g}(\sigma)<0$ for all metrics ${\mathrm g}$ with $\|{\mathrm g}-{\mathrm g}_0(t_*)\|_{C^{2,\alpha}}<\delta$. By the continuous dependence of Ricci flow on initial data @BGI20 [Thm A], there exist constants $\mathrm r>0$ and $C>0$, depending only on $t_*$ and ${\mathrm g}_0$, such that, if $\|{\mathrm g}_s-{\mathrm g}_0\|_{C^{4,\alpha}}\leq \mathrm r$, then $T({\mathrm g}_s)\geq t_0$ and $\|{\mathrm g}_s(t) - {\mathrm g}_0(t)\|_{C^{2,\alpha}} \leq C \|{\mathrm g}_s - {\mathrm g}_0\|_{C^{4,\alpha}}$ for all $t\in [0, t_0]$. Together with , this yields that if $0\leq s\leq \mathrm r/\lambda_{4,\alpha}$, then $$\|{\mathrm g}_s(t) - {\mathrm g}_0(t)\|_{C^{2,\alpha}}
 \leq C\,\|{\mathrm g}_s - {\mathrm g}_0\|_{C^{4,\alpha}} 
 \leq C\, \lambda_{4,\alpha}\, s, \quad \text{ for all } 0\leq t\leq t_0.$$ Thus, $\|{\mathrm g}_s(t_*)-{\mathrm g}_0(t_*)\|_{C^{2,\alpha}}<\delta$ and so $\sec_{{\mathrm g}_s(t_*)}(\sigma)<0$, for all $0\leq s<\delta/(C \,\lambda_{4,\alpha})$, while ${\mathrm g}_s={\mathrm g}_s(0)$ has $\sec>0$ if $s>0$ is sufficiently small, by .

\[rem:max-princ\] The curvature operators $R(t)\colon \wedge^2 TM\to\wedge^2 TM$ of metrics ${\mathrm g}(t)$ on $M^n$ evolving under Ricci flow satisfy the PDE $\frac{\partial}{\partial t}R=\Delta R+2Q(R)$, where $Q(R)$ depends quadratically on $R$. By Hamilton's Maximum Principle, if an $\mathsf O(n)$-invariant cone $\mathcal C\subset {\operatorname{Sym}}^2_{\mathrm b}(\wedge^2TM)$ is preserved by the ODE $\frac{{\mathrm{d}}}{{\mathrm{d}}t}R=2Q(R)$, then it is also preserved by the above PDE. It was previously known that the cone $\mathcal C_{\sec>0}$ of curvature operators with $\sec>0$ is *not* preserved under the above ODE on $R$ in dimensions $n\geq4$, since it is easy to find $R_0\in\partial \mathcal C_{\sec>0}$ with $Q(R_0)$ pointing outside of $\mathcal C_{\sec>0}$. Nevertheless, this observation alone does not imply the existence of metrics realizing such a family of curvature operators on some closed $n$-manifold, thus evolving under Ricci flow and losing $\sec>0$, as the above metrics ${\mathrm g}_s(t)$ do.

<span>BCRW19</span>

. . Ann. Global Anal. Geom., 50 (2016), 65–84.

. . Springer, Cham, 2015.

. . J. Geom. Anal., 30 (2020), 310–336.

. . Notices Amer. Math. Soc., 68 (2021), 1486–1498.

. . Invent. Math., 217 (2019), 95–126.

. . Comm. Anal. Geom., 27 (2019), 511–527.

. . SIAM J. Appl. Algebra Geom., 5 (2021), 200–228.

. . Math. Ann., 368 (2017), 971–986.

. . Geom. Funct. Anal., 17 (2007), 665–681.

. . Proc. Amer. Math. Soc., 143 (2015), 369–378.

. . Comment. Math. Helv., 9 (1936), 188–192.

. . Ann. of Math. (2), 152 (2000), 331–367.

. . Invent. Math., 149 (2002), 619–646.

. . J. Differential Geom., 17 (1982), 255–306.

. . J. Reine Angew. Math., 692 (2014), 153–171.

. , vol. 171 of Graduate Texts in Mathematics. Springer, Cham, third edition, 2016.

. . SIAM Rev., 49 (2007), 371–418.

. . In <span>*Proceedings of the <span>T</span>hirteenth <span>B</span>iennial <span>S</span>eminar of the <span>C</span>anadian <span>M</span>athematical <span>C</span>ongress (<span>D</span>alhousie <span>U</span>niv., <span>H</span>alifax, <span>N</span>.<span>S</span>., 1971), <span>V</span>ol. 2*</span>, 156–159. Canad. Math. Congr., Montreal, Que., 1972.

. . Transf. Groups, (2020). .

. . In <span>*Surveys in differential geometry. <span>V</span>ol. <span>XI</span>*</span>, vol. 11 of Surv. Differ. Geom., 25–62. Int. Press, Somerville, MA, 2007.

. . In <span>*Riemannian topology and geometric structures on manifolds*</span>, vol. 271 of Progr. Math., 233–262. Birkhäuser Boston, Boston, MA, 2009.

{'source': 'arXiv_out/2112'}