等参叶状结构的几何与拓扑

Differential geometry is an important part of fundamental research, while geometry and topology of submanifold is one of the core fields in differential geometry. We will devote ourselves to studying the geometry and topology of isoparametric foliations, which is currently one of the most vital research fields in submanifolds geometry. Since 2007, the mathematicians Cecil, Chi and Jensen, Immervoll, Miyaoka have published three papers in the authoritative journal 《Annals of Mathematics》. There are two major ingredients in this program. The first is the existence of isoparametric foliations on exotic manifolds, where we mainly consider the cases for exotic spheres and fake projective spaces, which playes a crucial role in understanding exotic differentiable structures . The second is about the isoparametric foliations on symmetric spaces, for which much attention is payed to the classification problem. Particularly, we will make an intensive study of the classification results of the isoparametric foliations on spheres and consider the related problems, such as geometry of double manifold and Bryant conjecture.

微分几何是基础研究的重要部分，而子流形的几何与拓扑是微分几何的核心领域之一。我们将致力于研究等参叶状结构的几何与拓扑。这个方向是目前国际上子流形几何研究的热点。自2007年来，相继有数学家Cecil，Chi，Jensen，Immervoll和Miyaoka在等参方面的研究成果共三篇文章发表在国际权威杂志《Ann. Math.》上。本项目主要研究怪异流形上等参结构的存在性和对称空间中的等参结构。首先，对于怪异流形上等参结构的研究，我们将主要考虑怪异球面和怪异投影空间的情形，这样的研究对于理解怪异微分结构有着重要的意义。其次，对于对称空间中的等参结构, 主要关心分类问题。特别地，我们将继续深入研究单位球面中等参结构的分类结果，进而考虑与之相关的double流形的几何和Bryant猜想。

本项目致力于研究等参叶状结构的几何与拓扑。首先，对于怪异流形上等参结构的研究，我们将主要考虑怪异球面和怪异投影空间的情形。这些的研究对于从微分几何的角度来理解怪异微分结构有着重要的意义。其次，我们深入地研究单位球面中等参结构的分类结果，进而考虑其在相关的几何问题中的应用，如特征值估计，调和映照，以及极小子流形的第二基本形的 pinching 问题。最后，我们研究等参叶状结构和曲率的关系，特别是对于 double流形的情形。