Ancient solutions of codimension two surfaces with curvature pinching in $\mathbb{R}^4$

Zhengchao Ji

doi:10.4134/BKMS.b190728

Bulletin of the
Korean Mathematical Society BKMS

ISSN(Print) 1015-8634 ISSN(Online) 2234-3016

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Bull. Korean Math. Soc. 2020; 57(4): 1049-1060

Online first article February 28, 2020 Printed July 31, 2020

https://doi.org/10.4134/BKMS.b190728

Ancient solutions of codimension two surfaces with curvature pinching in $\mathbb{R}^4$

Zhengchao Ji

Zhejiang University

Abstract

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Abstract

We prove rigidity theorems for ancient solutions of geometric flows of immersed submanifolds. Specifically, we find conditions on the second fundamental form that characterize the shrinking sphere among compact ancient solutions for the mean curvature flow in codimension two surfaces, which is different from the conditions of Risa and Sinestrari in \cite{RS} and we also remove the condition that the second fundamental form is uniformly bounded when $t\in(-\infty, -1)$.

Keywords: Mean curvature flow, ancient solutions, curvature pinching

MSC numbers: Primary 53C44, 35K55