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arXiv · 1609.09046

Uniqueness of grim hyperplanes for mean curvature flows

Ditter Tasayco·Detang Zhou
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Abstract

In this paper we show that an immersed nontrivial translating soliton for mean curvature flow in $\mathbb{R}^{n+1}$($n=2,3)$ is a grim hyperplane if and only if it is mean convex and has weighted total extrinsic curvature of at most quadratic growth. For an embedded translating soliton $Σ$ with nonnegative scalar curvature, we prove that if the mean curvature of $Σ$ does not change signs on each end, then $Σ$ must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.

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Ditter Tasayco, Detang Zhou. 2016-09-28. Uniqueness of grim hyperplanes for mean curvature flows. https://arxiv.org/abs/1609.09046

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