New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory
Li Lei, Hong-Wei Xu
备注：
This survey article was published in the Proceedings of the 7th International Congress of Chinese Mathematicians, Vol. I, 327-348, Adv. Lect. Math. 43, Int. Press, Somerville, MA, 2019
发表信息：
Proceedings of the 7th International Congress of Chinese Mathematicians, Vol. I, 327-348, Adv. Lect. Math. 43, Int. Press, Somerville, MA, 2019
In this survey, we will focus on the mean curvature flow theory with sphere theorems, and discuss the recent developments on the convergence theorems for the mean curvature flow of arbitrary codimension inspired by the Yau rigidity theory of submanifolds. Several new differentiable sphere theorems for submanifolds are obtained as consequences of the convergence theorems for the mean curvature flow. It should be emphasized that Theorem 4.1 is an optimal convergence theorem for the mean curvature flow of arbitrary codimension, which implies the first optimal differentiable sphere theorem for submanifolds with positive Ricci curvature. Finally, we present a list of unsolved problems in this area.
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The space of Fredholm operators of fixed index is stratified by submanifolds according to the dimension of the kernel. Geometric considerations often lead to questions about the intersections of concrete families of elliptic operators with these submanifolds: are the intersections non-empty? are they smooth? what are their codimensions? The purpose of this article is to develop tools to address these questions in equivariant situations. An important motivation for this work are transversality questions for multiple covers of $J$-holomorphic maps. As an application, we use our framework to give…
