Research of submanifolds and mean curvature flows in symmetric spaces by using the infinite dimensional geometry and the complexification

2016 Fiscal Year Final Research Report

• Project/Area Number: 25400076
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Geometry
• Research Institution: Tokyo University of Science
• Principal Investigator: Koike Naoyuki 東京理科大学, 理学部第一部数学科, 教授 (00281410)
• Project Period (FY): 2013-04-01 – 2017-03-31
• Keywords: 部分多様体幾何 / 無限次元幾何 / 平均曲率流 / リー群作用 / 対称空間 / 部分多様体の複素化

Outline of Final Research Achievements

Main results of this reseach are as follows. First we obtained the homogeneity theorem for ant-Kaehler isoparametric submanfolds in the infnite dimensional anti-Kaehler space under the assumption of a certain kind of diagonalzability of the shape operators and furthermore, completed almost the proof of the homogeneity theorem for certain kind of complex equifocal submanifolds in symmetric spaces of non-compact type by using the result. Secondly we obatined the collapsing theorem for the regularized mean curvature flow starting from horizontally convex invariant hypersurfaces in a Hilbert space. Thirdly we obtained a result about the volume-preserving mean curvature flow starting from tubes of nonconstant radius over certain reflective submanifold in rank one symmetric spaces of non-compact type.

Free Research Field

数物系科学