Abstract
In this paper we show that an immersed nontrivial translating soliton for a 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 \(\varSigma \) with nonnegative scalar curvature, we prove that if the mean curvature of \(\varSigma \) does not change signs on each end, then \(\varSigma \) must have positive scalar curvature unless it is either a hyperplane or a grim hyperplane.
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The second author was partially supported by CNPq and Faperj of Brazil.
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Tasayco, D., Zhou, D. Uniqueness of grim hyperplanes for mean curvature flows. Arch. Math. 109, 191–200 (2017). https://doi.org/10.1007/s00013-017-1057-9
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DOI: https://doi.org/10.1007/s00013-017-1057-9