The logarithmic Sobolev inequality on non-compact self-shrinkers

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 [10] 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 [21]. 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.