Pure and Applied Mathematics Quarterly

Volume 12 (2016)

Number 4

A global pinching theorem for complete translating solitons of mean curvature flow

Pages: 603 – 619

DOI: https://dx.doi.org/10.4310/PAMQ.2016.v12.n4.a8

Authors

Huijuan Wang (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Hongwei Xu (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Entao Zhao (Center of Mathematical Sciences, Zhejiang University, Hangzhou, China)

Abstract

In the present paper, we prove that for a smooth complete translating soliton $M^n (n \geq 3)$ with the mean curvature vector $H$ satisfying $H = V^N$ for a unit constant vector $V$ in the Euclidean space $\mathbb{R}^{n+p}$, if the trace-free second fundamental form $\mathring{A}$ satisfies $(\int_M {\lvert \mathring{A} \rvert}^n d \mu)^{1/n} \lt K(n), \int_M {\lvert \mathring{A} \rvert}^n e^{\langle V, X \rangle} d \mu \lt \infty$, where $K(n)$ is an explicit positive constant depending only on $n$, then $M$ is a linear subspace.

Keywords

rigidity theorem, translating soliton, integral curvature pinching

2010 Mathematics Subject Classification

53C42, 53C44

Received 30 January 2018

Published 26 July 2018