A Bernstein type theorem for ancient solutions to the mean curvature flow in arbitrary codimension

Guan, Li; Xu, Hongwei; Zhao, Entao

doi:10.1090/proc/16078

A Bernstein type theorem for ancient solutions to the mean curvature flow in arbitrary codimension
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by Li Guan, Hongwei Xu and Entao Zhao

Proc. Amer. Math. Soc. 151 (2023), 269-279

DOI: https://doi.org/10.1090/proc/16078

Published electronically: September 2, 2022

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Abstract:

We prove a Bernstein type theorem for ancient solutions to the mean curvature flow in higher codimension. More precisely, we show that for a complete ancient solution $\{M_t\}_{t\in (-\infty ,0)}$, if the mean curvature vector is uniformly bounded and the $\omega$-function for $M_t$ satisfies $\omega \geq \omega _0$ uniformly for some constant $\omega _0>\frac {1}{\sqrt {2}}$, then $M_t$ must be an affine subspace for each $t\in (-\infty ,0)$.

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