The 2nd Shot of The 13th MSJ-SI“Diﬀerential Geometry and Integrable Systems”
The 5th International Workshop
Geometry of Submanifolds and Integrable Systems
November 26 - December 1, 2022
Abstracts
Plenary Speakers
November 26.
Shu-Cheng Chang (National Taiwan University, Taiwan)
Title: The Legendrian Mean Curvature Flow
Abstract
It is due to Colin-Giroux-Honda that, for ﬁxing a Thurston–Bennequin invariant, a rotation
number, and a knot type in contact 3-manifold, there are only ﬁnitely many Legendrian knot
types realizing this data. However, it is proved by Ekholm-Etnyre-Sullivan that there is an inﬁnite family of Legendrian embeddings of the n-sphere and n-torus into the standard contact
Euclidean (2n + 1)-space, n > 1 that are not Legendrian isotopic even though they have the samerotation number and Thurston-Bennequin invariant. On the other hand, it is closely related
between Legendrian and Lagrangian submanifolds. More precisely, the projection of the mean
curvature of a Legendrian submanifold in the contactization of a Kaehler manifold coincides withthe mean curvature of the projected Lagrangian submanifold in such a Kaehler manifold. In particular, the projection of an minimal Legendrian submanifold is an immersed minimal Lagrangiansubmanifold. In this lecture, we will address the recent result on the isotopic Legendrian problem and existence of minimal Legendrian submanifolds in η-Einstein Sasakian manifolds via the
Legendrian mean curvature ﬂow which is initiated by K. Smoczyk. Parts of the talk is based on
the jointed work with Y. Han and C.-T. Wu.

Ryosuke Takahashi (Kyushu University, Japan)
Title: A Nakai-Moishezon type criterion for supercritical deformed Hermitian-Yang-Mills
equation
Abstract
In this talk, we show that the solvability of the supercritical deformed Hermitian-Yang-Mills
equation is equivalent to a certain algebraic positivity condition modeled on the Nakai-Moishezon
ampleness criterion, which conﬁrms the mirror version of the Thomas-Yau conjecture. This is a
joint work with J. Chu (Peking University) and M.-C. Lee (Chinese University of Hong Kong).

Miyuki Koiso (Kyushu University, Japan)
Title: A variational problem for developable surfaces with curved folds
Abstract
Developable surfaces are surfaces which can be unfolded into the plane preserving the lengthof all curves on the surface. It is known that smooth developable surfaces in the three-dimensional
Euclidean space are surfaces with zero Gaussian curvature, and they are generalized cylinders,
generalized cones (away from apexes), and tangent developable surfaces. Since developable surfaces with curved folds are constructed by bending a ﬂat sheet, they have many applications in
manufacturing objects. In this talk, we discuss a variational problem for ”pillow boxes” which
are a speciﬁc class of closed developable surfaces with genus zero and with curved folds. For agiven initial closed surface in a plane which consists of two rectangles with the same size whoseboundaries are identiﬁed, we prove the existence and uniqueness of the maximizer of the enclosed
volume (called the optimum solution) among piecewise smooth “cylindrical surfaces” which areisometric to the initial surface. Moreover, we give the explicit representation formula for the
optimum solution, and show that there exists an isometric deformation from the initial planar
surface to the optimum solution. (Joint work with Kento Okuda.)

November 27.
Pak Tung Ho (Sogang Univeristy, Korea)
Title: The weighted Yamabe problem
Abstract
In this talk, I will explain what the weighted Yamabe problem is, and mention some related
results that Jinwoo Shin (KIAS) and I obtained.

Hiroyasu Sato (Nippon Institute of Technology, Japan)
Title: The horosphere version of the Osserman conjecture and related topics
Abstract
The horosphere version of the Osserman conjecture is stated as follows. Let M be a simply
connected, complete manifold with no focal points. If every horosphere has constant principal
curvatures (with multiplicities) and these principal curvatures (with multiplicities) do not depend
on a choice of horospheres in M , then M is a rank-one symmetric space of non-compact type ora ﬂat Euclidean space. In this talk, we introduce the above conjecture and related results. This
work is based on the joint work with M. Itoh.

November 29.
Indranil Biswas (Tata Institute of Fundamental Research, India)
Title: Moduli spaces of framed logarithmic and parabolic connections on a Riemann surface
Abstract
We construct the moduli spaces of framed logarithmic connections and also framed parabolic
connections on a Riemann surface with marked points. It is proved that these moduli spaces
have algebraic symplectic structure. We also compute the degree of transcendence of the global
algebraic functions on the moduli spaces. This is a joint work with Michi-aki Inaba, Arata Komyo
and Masa-Hiko Saito.

Charles Ouyang (UMass Amherst, USA)
Title: Compactiﬁcations of Hitchin components
Abstract
Hitchin components are natural generalizations of the classical Teichmüller space. In the setting of SL(3, R), the Hitchin component parameterizes the holonomies of convex real projective
structures, which are related to hyperbolic aﬃne spheres. By studying Blaschke metrics, which
are Riemannian metrics associated to hyperbolic aﬃne spheres, along with their limits, we obtain a compactiﬁcation of the SL(3, R) Hitchin component. We show the boundary objects are
hybrid structures, which are in part ﬂat metric and in part laminar. These hybrid objects arenatural generalizations of measured laminations, which are the boundary objects in Thurston’scompactiﬁcation of Teichmüller space. Time permitting, we discuss the other rank 2 settings of
PSL(2, R) × PSL(2, R), Sp(4, R) and G2 .

Yuuki Tadokoro (National Institute of Technology, Kisarazu College, Japan)
Title: Nonlinear O(3) sigma model in discrete complex analysis
Abstract
Mercat introduced a discrete Riemann surface with a discrete complex structure on bipartite
quad-graphs on surfaces, and gave an algorithm to compute its period matrix. Bobenko andGünther treated Mercat’s discrete Riemann surfaces more precisely and deﬁned a new discrete
complex structure using the medial graphs of the above quad-graphs.
We consider discrete Riemann surfaces on planar quad-graphs and its O(3) sigma model.
We deﬁne the weighted discrete Dirichlet energy and area obtained from this sigma model, andexplain the inequality between them. This is a discrete version of the inequality of Belavin andPolyakov. For general quad-graphs with orthogonal diagonals, we also show that the discrete (anti)holomorphic function satisﬁes the Euler-Lagrange equations derived from the discrete energy.
On some special rhombic lattices, the discrete area can be regarded as the mapping degree ofthe discrete power function. Furthermore, we show that the discrete energy, discrete area, andEuler-Lagrange equations tend to their continuous counterparts as the lattice spacing approaches
zero.
This talk is based on joint work with Masaru Kamata and Masayoshi Sekiguchi.

Wayne Rossman (Kobe University, Japan)
Title: Discrete Omega surfaces and their transformations
Abstract
This talk lies within the ﬁeld of discrete diﬀerential geometry (DDG), and will focus on
recent joint work with Fran Burstall, Joseph Cho, Udo Hertrich-Jeromin and Mason Pember on
discretization of Omega surfaces.
Such surfaces, in both the smooth and discrete settings, are a generalization of isothermic
surfaces when viewed as Legendre immersions in 3-dimensional spaceforms (i.e. Euclidean, spherical and hyperbolic 3-spaces), as they have isothermic sphere congruences although they might
not be isothermic themselves. Because of this, they have an underlying integrable-systems-based
mathematical structure, and a resulting transformation theory.
Using Moebius and Lie sphere geometries, we will describe the equivalence of a number ofapproaches to discrete Omega surfaces, and also a satisfying transformation theory for them.

November 30.
Robert Kusner (Online, UMass Amherst, USA)
Title: Steklov Eigenspaces and Free Boundary Minimal Surfaces
Abstract
The coordinate functions on a free boundary minimal surface (FBMS) in the unit ball are
Steklov eigenfunctions with eigenvalue 1. For many embedded FBMS in B 3 we show its ﬁrst
Steklov eigenspace coincides with the span of its coordinate functions, aﬃrming a conjecture of
Fraser & Li in an even stronger form.

Aya Ishizeki, Takeyuki Nagasawa (Saitama University, Japan)
Title: Möbius energy for knots and links, and wave maps
Abstract
The Möbius energy for knots can be decomposed into three parts. The ﬁrst two are functionals
which are still invariant under Möbius transformations, and the third one is an absolute constant.
A similar decomposition also holds for the Möbius energy for links. The second part has amore complicated expression than the ﬁrst one. Recently, the authors found its new and simple
expression by using the Gauss map. Regarding it as a functional for maps from the 2-dimensional
torus with Lorenz metric into a sphere, we observe that it is the energy of harmonic maps. Thatis, critical maps are wave maps. Here we consider knots and links which Gauss maps are wavemaps.

Mao-Pei Tsui (National Taiwan University, Taiwan)
Title: A new monotone quantity in mean curvature ﬂow implying sharp homotopic criteria
Abstract
We discuss a new monotone quantity in graphical mean curvature ﬂows of higher codimensions.
The submanifold deformed by the mean curvature ﬂow is the graph of a map between Riemannian
manifolds, and the quantity is monotone increasing under the area decreasing condition andsuitable ambient curvature condition. The ﬂow provides a natural homotopy of the corresponding
map and leads to sharp criteria regarding the homotopic class of maps between complex projective
spaces, and maps from spheres to complex projective spaces, among others. This is joint workwith Chung-Jun Tsai and Mu-Tao Wang.

Takashi Sakai (Tokyo Metropolitan University, Japan)
Title: Variational problems for integral invariants of the second fundamental form of a map
between Riemannian manifolds
Abstract
Motivated by integral geometry, we deﬁne integral invariants of a map between Riemannian
manifolds by using invariant functions of the second fundamental form. Then we discuss variational problems for these integral invariants. We derive the ﬁrst variational formulae for integral
invariants deﬁned from invariant homogeneous polynomials of degree two. Among these integral
invariants, we show that the Euler–Lagrange equation of the Chern–Federer energy functional isreduced to a second order PDE. Then we give some examples of Chern–Federer submanifolds inRiemannian space forms. This talk is based on a joint work with Rika Akiyama and Yuichiro
Sato.

Reiko Miyaoka (Tohoku University, Japan)
Title: Various aspects of isoparametric hypersurfaces
Abstract
In this talk, I will give various aspects of isoparametric hypersurfaces. Starting from the origin
in geometric optics, I introduce isoparametric hypersurfaces, and state the classiﬁcation. Then we
consider the Gauss images of the hypersurfaces in the complex hyperquadric, and give a partial
answer to the Arnold-Givental conjecture. As another aspect, the isoparametric functions are
related to analytic problems such as the Pompeiu problem and the Schiﬀer problem. I mention
recent informations on these problems.

December 1.
Emma Carberry (The University of Sydney, Australia)
Title: Constant Mean Curvature Tori in R3 of Spectral Genus Two
Abstract
A constant mean curvature (cmc) torus in R3 can be described in terms of a hyperelliptic
spectral curve, whose genus g is at least two. The countably inﬁnite space of examples studied
by Wente and Abresch are contained in this simplest case of g = 2, but there are many more
cmc tori in this simplest class. The object of this talk is to give a full account of the space of all
cmc tori that have a spectral curve of genus g = 2. I shall describe how Whitham deformations
can be used to give a complete account of the space of spectral data of real solutions of thesinh–Gordon equation of spectral genus 2. More precisely, I shall give a parameterisation of theclosure of spectral data of constant mean curvature tori in R3 and analyse the boundary of thisset. As a special case the Wente family, which is described by spectral data with real coeﬃcients,
is 1–dimensional, smooth and connected.
This is joint work with Martin Kilian, Sebastian Klein and Martin Schmidt.

Franz Pedit (UMass Amherst, USA)
Title: Higgs bundles, Monge-Ampere equations, aﬃne spheres, and SYZ geometry
Abstract
Strominger-Yau-Zaslow suggest that Mirror Symmetry can be seen as T-duality on CalabiYau 3-folds ﬁbered by special Lagrangian 3-tori which degenerate along a singular set modelled
on a trivalent graph. We shall construct inﬁnitely many such ﬁbrations, singular over a Yvertex, using the parabolic non-Abelian Hodge correspondence relating Higgs bundles, surface
group representations, and equivariant harmonic maps. Along the way, we shall encounter rather
classical geometric objects, the aﬃne spheres introduced by the Blaschke school in the 1930s.

Short Communications (Parallel Session)
November 27.
Ryo Takenaka (Osaka Metropolitan University, Japan)
Title: Fermionic character formulas of aﬃne Lie algebras and related topics
Abstract
Consider twisted aﬃne Lie algebras obtained from simple Lie algebras and their automorphisms. The standard module is the integrable highest weight module. By using the vertex
operator construction, we were able to obtain combinatorial bases of standard modules and somerelated algebras for twisted aﬃne Lie algebras. In this talk, we introduce these results and possibility of application for several topics.

Shota Hamanaka (Mitsubishi Electric Corporation Advanced Technology R&D Center, Japan)
Title: C 0 , C 1 limit theorems for total scalar curvatures
Abstract
We give some C 0 or C 1 limit theorems for total scalar curvatures. More precisely, we showthat the lower bound of the total scalar curvatures on a closed manifold is preserved under theC 0 or C 1 convergence of the Riemannian metrics under some assumptions. Moreover, we givesome counterexamples to the above theorems on an open manifold.

Yuki Koto (Kyoto University, Japan)
Title: Analytic decomposition of quantum cohomology
Abstract
Quantum cohomology of a smooth projective variety X is a formal deformation of the ordinary
cohomology of X, and plays a very signiﬁcant role in mirror symmetry.
Since QH is not functorial, it is interesting to investigate a relationship of quantum cohomology
of diﬀerent varieties. In this talk, we discuss quantum cohomology of a ﬁber bundle E over B
whose ﬁber is a toric variety. We observe that, under a suitable assumption, quantum cohomology(D-module) of E is analytically decomposed into copies of that of B.

Homare Tadano (Yamaguchi University, Japan)
Title: A Zoo of Bonnet-Myers Type Theorems
Abstract
One of important issues in diﬀerential geometry is to investigate the relation between curvatureand topology of Riemannian manifolds. I will introduce various Bonnet-Myers type theorems forcomplete Riemannian manifolds via m-Bakry-Émery Ricci curvature.

Denis Polly (Kobe University, Japan)
Title: Representations of discrete Bryant type surfaces
Abstract
The famed Weierstrass-Enneper representation for minimal surfaces in Euclidean space hasan analogue in hyperbolic space: originally introduced for cmc 1 surfaces by Bryant in 1987 it
can be used to generate a class of linear Weingarten surfaces called Bryant-type surfaces. This
class covers the aforementioned cmc 1 surfaces but also ﬂat fronts as well as constant harmonicmean curvature 1 surfaces.
In the realm of integrable discrete diﬀerential geometry, Bryant-type surfaces can be deﬁned
either via a discrete version of their representation or via discrete analogues of the linear Weingarten condition their smooth counterparts satisfy. In special instances it has been shown thatthese deﬁnitions coincide.
In this talk we provide an interpretation of the discrete Weierstrass-type representation for
Bryant-type surfaces in terms of the transformation theory of discrete Omega surfaces. Further,
we will prove that all Bryant-type surfaces, as deﬁned via their linear Weingarten condition, canbe generated via this representation.

Kuang-Ru Wu (Academia Sinica, Taiwan)
Title: Positively curved Finsler metrics on vector bundles
Abstract
While the equivalence between ampleness and positivity holds for vector bundles of rank one,its higher rank counterpart known as Griﬃths’ conjecture is still open. There is also a similar but
weaker conjecture by Kobayashi who proposed to use Finsler rather than Hermitian metrics tostudy the equivalence. We will review these two conjectures and state our progress. One of ourresults is that Kobayashi positivity implies ampleness and convex Kobayashi positivity. We will
also discuss how to prove Kobayashi positivity for ample vector bundles with additional curvatureassumptions.

Joseph Cho (TU Wien, Austria)
Title: Closed discrete Darboux transforms
Abstract
In this talk, we discuss the conditions to obtain closed Darboux transforms of discrete isothermic surfaces. We test the robustness of our theory by creating concrete examples. This talk isbased on the joint work with Katrin Leschke (University of Leicester) and Yuta Ogata (Kyoto
Sangyo University).

Brian Harvie (National Taiwan University, Taiwan)
Title: The Bartnik quasi-local mass of small geodesics balls
Abstract
A fundamental problem in Einstein’s theory of general relativity is to ﬁnd a suitable notionof the mass of a space-like, time-symmetric hypersurface in a spacetime. The Arnowitt-DeserMisner (ADM) mass manages to measure the total mass of such a slice in an isolated gravitating
system, but determining the mass contained within a compact region via a geometric quantity ismore diﬃcult. R. Bartnik proposed a deﬁnition for the mass of a compact domain that utilizes
the ADM mass and has an important relationship with equilibrium solutions to the Einsteinequations called ”static” spacetimes.
In this talk, I will discuss recent work with Prof. Ye-Kai Wang about estimating the Bartnik
mass of small geodesic balls in Riemannian 3-manifolds by studying their static extensions, whichbuilds upon earlier work by David Wiygul. I would like to make the talk accessible to diﬀerential
geometers without a relativity background, and so I will focus mostly on exposition.

Artur Sergyeyev (Online, Silesian University in Opava, Czech Republic)
Title: From contact geometry to multidimensional integrable systems
Abstract
In this talk we showcase a novel application of three-dimensional contact geometry, where it
helps answering a longstanding question of just how exceptional are partial diﬀerential systems
in four independent variables that are integrable in the sense of soliton theory. It turns out thatsuch systems are far more numerous than it was believed, and we provide an eﬀective explicit
construction, involving contact vector ﬁelds, for a large class of systems in question along with
their Lax pairs. As a byproduct, we present a ﬁrst example of an integrable partial diﬀerential
system in four independent variables with a nonisospectral Lax pair which is algebraic, rather
than rational, in the spectral parameter.

Tetsuya Nakamura (UMass Amherst)
Title: Sp(C 4 ) Transformations of CMC 1 Surfaces in H 3
Abstract
Constant mean curvature (CMC) 1 surfaces in 3 dimensional hyperbolic space H 3 are represented by a null curve immersion into a 3 dimensional quadric in CP 4 . This representation is due
to Bryant. Linear group SO(C 5 ), which leaves the quadric invariant, naturally acts on null curve
immersions. In this talk, we will see with some examples that this action gives rise to (locallydeﬁned) non-trivial deformations of the corresponding CMC 1 surface in H 3 . Furthermore thestandard action of its spin group Sp(C 4 ) on C 4 is understood as a linear transformation of spinors
associated with the CMC 1 surface. From this perspective, we will observe that the transformation is globally deﬁned if we consider a special class of CMC 1 surfaces, namely of genus 0 andwith smooth ends. In addition to it, the transformation preserves the Willmore energy of thesurface.

November 28.
Ye-Kai Wang (Online, National Yang Ming Chiao Tung University, Taiwan)
Title: A rigidity result for surfaces in Schwarzschild spacetime
Abstract
We study spacelike surfaces in Schwarzschild spacetime whose mean curvature vector hasconstant norm. When the surface lies in the standard null hypersurface, the problem is closely
related to the Liouville and Obata Theorem in conformal geometry. We show that such surfaces
must lie in a time slice. The talk is based on the joint work with Po-Ning Chen at UC Riverside.

Albert Wood (National Taiwan University, Taiwan)
Title: Cohomogeneity-One Lagrangian Mean Curvature Flow
Abstract
Lagrangian mean curvature ﬂow is the name given to the observation that Lagrangian submanifolds of Calabi-Yau manifolds are preserved under mean curvature ﬂow. This observation
gave rise to a conjecture of Thomas-Yau, which states that assuming a stability condition on
the Lagrangian, the ﬂow should converge to a unique volume-minimising representative: a special
Lagrangian. Since mean curvature ﬂow typically forms ﬁnite-time singularities, a surgery procedure must be deﬁned to resolve the conjecture, and an understanding of the possible singularity
models is a vital ﬁrst step.
In this work, we study Lagrangians that are invariant with respect to a group action respecting
the Calabi-Yau structure with (n − 1)-dimensional orbits. Such Lagrangians must be containedin an (n + 1)-dimensional submanifold, a level set of the moment map, and taking the symplectic
quotient produces a curve in a 2-manifold. Lagrangian mean curvature ﬂow may therefore be
studied via a related curve shortening ﬂow, which we show does not depend on the group action.
By this method, we are able to classify cohomogeneity-one shrinking and expanding solitons, aswell as fully classify singularities in the case of the zero level set.

Taiki Yamada (Shimane University, Japan)
Title: Asymmetric allocation rule depending on graph structure
Abstract
This talk will be given in collaboration with Taisuke Matsubae (Chuo University), and will bebased on arXiv:2110.06506. The primary focus in cooperative game theory concerns the sharing
or distribution of the worth gained from a cooperation. To this end, we usually consider Shapley
value that introduced in 1953. However, since this value does not reﬂect the structure of thegraph, we deﬁne a new allocation rule by using the Shapley value. In this talk, we will compare
the Shapley values with our allocation rule and also present its relevance to discrete diﬀerential
geometry.

Yen-Chang Huang (National University of Tainan, Taiwan)
Title: Some integral geometric formulas in the Heisenberg groups
Abstract
In this talk, I will introduce our recent works related to some integral formulas in Cauchy
Riemann manifolds (CR manifolds). The classical Cauchy surface area formula sates that thesurface area of any compact hypersurface in the n-dimensional Euclidean space equals the averageof all projected surface areas onto the (n − 1)-subspaces; the similar result can be generalized
to the three-dimensional Heisenberg group, which is regard as a one-dimensional CR manifoldwith zero Tanaka-Webster curvature. In contract to the Euclidean case, the properties of thePansu spheres play the key ingredient in the proofs instead of the round spheres. The result
supports that the Pansu spheres may be a more appropriate model in studying the geometry of
the Heisenberg groups rather than the round spheres.

Chih-Wei Chen (National Sun Yat-sen University, Taiwan)
Title: Hessian estimators on data manifolds
Abstract
Given a function deﬁned on a data set, there have been several ways to estimate the Hessianof it. Although the estimators can be quite accurate, their constructions are all based on certain
grid structures. Without using any grid assumption, we construct a Hessian estimator whichconverges to the continuous Hessian tensor when the data points are densely sampled from a data
manifold. This is a joint work with Dr. Hau-Tieng Wu.

Abhitosh Upadhyay (Indian Institute of Technology Goa, India)
Title: On compact embedded Weingarten hypersurfaces in warped products
Abstract
The well-known Alexandrov theorem ensures that a closed embedded hypersurface of the
Euclidean space with constant mean curvature must be a round sphere. It was proved by Montiel
and Ros that the Alexandrov theorem for mean curvature as well as higher order mean curvatures
is also true for hypersurfaces of hyperbolic spaces and half-spheres. Recently, de Lima proved acomparable result for the so called linear Weingarten hypersurfaces satisfying Hr = aH + b for
two real constants a ≥ 0 and b > 0, where H and Hr are respectively the mean curvature andthe r-th mean curvature of the hypersurfaces. In this talk, I will discuss that Lima’s result also
holds for a large class of warped products which contains in particular the hyperbolic spaces andthe half-spheres.