New Developments in Mean Curvature Flow of Arbitrary Codimension Inspired By Yau Rigidity Theory

Abstract

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.

arXiv commentThis 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

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• Normal scalar curvature conjecture and its applications
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Cited by in corpus (4)

• Ancient Solution of Mean Curvature Flow in Space Forms
• Surfaces pinched by normal curvature for mean curvature flow in space forms
• On the mean curvature flow of submanifolds in the standard Gaussian space ${}^{†}$
• A sharp convergence theorem for the mean curvature flow in spheres I