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arXiv · 2410.13601

The Logarithmic Sobolev inequality on non-compact self-shrinkers

Guofang Wang·Chao Xia·Xiqiang Zhang
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Abstract

In the paper we establish an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which generalizes a recent result of Brendle \cite{Brendle22} for closed self-shrinkers. We first provide a proof for the logarithmic Sobolev inequality in the Euclidean space by using the Alexandrov-Bakelman-Pucci (ABP) method. Then we use this approach to show an optimal logarithmic Sobolev inequality for complete, non-compact, properly embedded self-shrinkers in the Euclidean space, which is a sharp version of the result of Ecker in \cite{Ecker}. The proof is a noncompact modification of Brendle's proof for closed submanifolds and has a big potential to provide new inequalities in noncompact manifolds.

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Guofang Wang, Chao Xia, Xiqiang Zhang. 2024-10-17. The Logarithmic Sobolev inequality on non-compact self-shrinkers. https://arxiv.org/abs/2410.13601

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