Submanifold Geometry, Lie Group Action and Its Applications to Theoretical Physics 2025

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Xiaobo Liu (Peking University)
Juncheol Pyo (Pusan National University)
Martin Guest (Waseda University, NCTS (=National Center for Theoretical Science), Taiwan): online
Kurando Baba (Tokyo University of Science)
Tomoki Fujii (Tokyo University of Science)
Shota Hamanaka (Osaka University)
Osamu Ikawa (Kyoto Institute of Technology)
Isami Koga (Kyushu International University)
Yuuki Sasaki (Utsunomiya University)
Yuichiro Sato (Waseda University)
Masahiro Morimoto (Tokyo Metropolitan University)
Yuta Yamauchi (Yokohama National University)

Xiaobo Liu (Peking University)
Title: Mean Curvature Flow for Isoparametric Submanifolds in Hyperbolic Spaces
Abstract: Mean curvature flow (MCF) of isoparametric submanifolds in Euclidean spaces and spheres have been studied by Liu and Terng. They have been used to give explicit ancient solutions of MCF with complicated topological type and study rigidity of ancient solutions of MCF for hypersurfaces in spheres. In this talk, I will describe behavior of MCF of isoparametric submanifolds in hyperbolic spaces. This talk is based on joint work with Wanxu Yang.

Juncheol Pyo (Pusan National University)
Title: Translating Solitons for the Mean Curvature Flow
Abstract: Translating solitons and self-shrinkers are solitons for the mean curvature flow (MCF). They serve not only as blow-up models for MCF singularities but also as minimal surfaces in certain Riemannian manifolds. In this talk, we compare properties of minimal surfaces and MCF solitons, particularly with respect to Bernstein-type theorems and properness. More precisely, we present rigidity results for graphical translators that move in nonvertical directions. In addition, we introduce sufficient conditions for the properness of translating solitons. This is based on joint work with Daehwan Kim, Yuan Shyong Ooi, and John Ma.

Martin Guest (Waseda University, NCTS, Taiwan)
Title: A symplectic manifold arising in the theory of meromorphic connections
Abstract: Adjoint orbits of Lie groups are well known examples of symplectic manifolds which have various roles in submanifold theory, Lie theory/representation theory, and quantization. We discuss a new example which seems to share these characteristics. It arises naturally as a space of monodromy data which parametrizes certain meromorphic connections; in this sense it is an example of a moduli spaces of flat connections, as in the theory of Painleve equations or harmonic bundles. On the other hand it has a purely Lie-theoretic description, as a submanifold of the universal centralizer. It also has the structure of a symplectic groupoid, which makes it a subspace of the symplectic groupoid constructed by Bondal in the theory of triangulated categories.

Kurando Baba (Tokyo University of Science)
Title: On Backward Mean Curvature Flow for Equifocal Submanifolds in Symmetric Spaces of Compact Type
Abstract: It is known that the backward mean curvature flows starting from equifocal submanifolds have long-time solutions and converge to minimal equifocal submanifolds. In this talk, we give observations of these long-time solutions in terms of the tangential focal datum associated with the initial equifocal submanifolds. This talk is based on joint work with Naoyuki Koike.

Tomoki Fujii (Tokyo University of Science)
Title: Shapes of graphical solitons for the mean curvature flow invariant under hyperpolar actions
In this talk, we discuss solitons for the mean curvature flow given by graphs of functions invariant under actions on symmetric spaces. First, we introduce the classification of shapes of graphical translating solitons invariant under isotropy actions or Hermann actions on symmetric spaces. Next, we show the recent works on shapes of graphical rotating solitons invariant under actions of cohomogeneity one. This talk is based on joint work with Naoyuki Koike (Tokyo University of Science).

Shota Hamanaka (Osaka University)
Title: Convergence rate of geometric flows on weighted Riemannian manifolds
Abstract: ：In my talk, I will discuss the convergence rate of geometric flows on weighted Riemannian manifolds (i.e., Riemannian manifold with weighted measure). This is based on a joint work with Pak Tung Ho (Tamkang University). Carloto-Chodosh-Rubinstein studied the convergence rate of the Yamabe flow and discovered that there is a Riemannian manifold to which the flow converges exactly at a polynomial rate. In this talk, I will show that such a phenomenon also occurs on the more general space, weighted Riemannian manifold.

Osamu Ikawa (Kyoto Institute of Technology)
Title: The intersection of two real flag manifolds in a complex flag manifold -An application of a canonical form of a compact symmetric triad-
Abstract: We state necessary and sufficient conditions for the intersection of two real flag manifolds in a complex flag manifold to be discrete, including previously known results. A discrete intersection of real flag manifolds is an orbit of a Weyl group and becomes the antipodal set of the complex flag manifold. A complex flag manifold and a pair of real flag manifolds are constructed from a compact symmetric triad, and we show that the compact symmetric triad can be chosen to be canonical. Under the setting, we show some properties of the intersection. We discuss the future work in the last of the presentation.
This talk is based on joint work with Kurando Baba and Shinji Ohno. It also contains the contents of joint work with Hiroshi Iriyeh, Takayuki Okuda, Takashi Sakai, and Hiroyuki Tasaki.

Isami Koga (Kyushu International University)
Title: Equivariant harmonic maps of the quaternionic projective spaces into Grassmann manifolds
Abstract: In this talk, I will introduce rigidity theorems of Sp(m+1) equivariant harmonic maps of the quaternionic projective spaces of dimension m into Grassmann manifolds. At first we consider the case where the target manifold is the complex Grassmannians. Next we also consider the case where the target is real or quaternionic Grassmannians.To show them we use a generalized do Carmo-Wallach theory, which is based on a generalization of a Theorem of Tsunero Takahashi. This talk is based on a joint work with Yasuyuki Nagatomo (Meiji University) and Masaro Takahashi (National  Institute of Technology, Kurume college).

Yuuki Sasaki (Utsunomiya University)
Title: Maximal antipodal sets of exceptional symmetric spaces
Abstract: In symmetric spaces, a finite discrete set called an antipodal set is defined. While it has been pointed out that antipodal sets are related to various mathematical structures on symmetric spaces, there still exist symmetric spaces for which the classification of maximal antipodal sets remains incomplete. In this talk, I will present classification results of maximal antipodal sets in exceptional　compact symmetric spaces. In particular, I will show that maximal antipodal sets in each exceptional compact symmetric space can be constructed using either the octonions, a maximal torus, or the Weyl group. If time permits, I will also introduce inclusion relations among exceptional compact symmetric spaces that were discovered through the study of antipodal sets.

Yuichiro Sato (Waseda University)
Title: Construction of higher-dimensional vacuum solutions using almost abelian Lie groups
Abstract: We investigate spatially homogeneous or homogeneous spacetimes in arbitrary dimensions, and construct vacuum solutions of the Einstein equations without a cosmological constant. An almost abelian Lie group is defined as a Lie group whose Lie algebra admits a codimension-one abelian ideal. Assuming that the spatial part of a spatially homogeneous spacetime, or the whole spacetime itself, is modeled on an almost abelian Lie group, we derive the Ricci-flat conditions for such spacetimes. In particular, we generalize the classical four-dimensional Taub and Petrov solutions to higher dimensions. Moreover, we show that in the time evolution of these solutions, the spatial dimensions cannot expand or contract simultaneously in the late-time limit. This talk is based on joint work with Takanao Tsuyuki (Hokkaido Information University).

Masahiro Morimoto (Tokyo Metropolitan University)
Title: Affine differential geometry of parallel transport maps and weakly reflective submanifolds
Abstract: In the 1990s, C.-L. Terng and G. Thorbergsson investigated a natural Riemannian submersion from an infinite dimensional Hilbert space onto a compact Riemannian symmetric space G/K. This map is called the parallel transport map over G/K. Afterward, N. Koike extended their theory to the case that G/K is a Riemannian symmetric space of non-compact type. In this talk, I will explain my research on the extension of those theories in the framework of affine differential geometry. In particular, the parallel transport map over an affine symmetric space is defined and shown to be an affine submersion with horizontal distribution in the sense of Abe and Hasegawa. Its relation to weakly reflective submanifolds in symmetric spaces will also be discussed.

Yuta Yamauchi (Yokohama National University)
Title: The total absolute curvature of submanifolds with singularities
Abstract: For an $n$-dimensional immersed compact submanifold in Euclidean space $\mathbb{R}^{n+r}$, it is known that the total absolute curvature is greater than or equal to the sum of the Betti numbers. Moreover, the total absolute curvature is equal to $2$ if and only if the submanifold is a convex hypersurface embedded in an affine $(n+1)$-subspace of $\mathbb{R}^{n+r}$ (the Chern-Lashof theorem). In this talk, we show a Chern-Lashof type theorem for submanifolds with singularities in Euclidean space. More precisely, we prove that for an $n$-dimensional admissible compact frontal in $\mathbb{R}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$ and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-subspace of $\mathbb{R}^{n+r}$.