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)
Kurando Baba (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)

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.

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).