几何变分问题近期发展

Recent progress on geometric variational theory

本文回顾与平均曲率相关的几何变分理论的最新研究进展,主要讨论极小超曲面的Morse理论,并重点介绍常平均曲率(constant mean curvature,CMC)曲面的变分理论、重数一猜想、极小超曲面的典范空间分布、自由边界极小曲面的变分理论及在等变情形下的推广.

In this paper,we survey recent progress on the variational theory related to the area functional and the mean curvature.We mainly discuss the Morse theory of minimal hypersurfaces.In particular,we focus on the variational theory for constant mean curvature(CMC)surfaces,the multiplicity one conjecture,generic spatial distributions of minimal hypersurfaces,the variational theory for minimal surfaces with free boundary,and generalizations in the equivariant settings.