Sobolev Inequalities in Spacelike Submanifolds of Minkowski Space

Abstract

We follow the method of ABP estimate in \cite{brendle2021} and apply it to spacelike submanifolds in ${\mathbb{R}}^{n,1}$. We then obtain Michael-Simon type inequalities. Surprisingly, our investigation leads to a Sobolev inequality without a mean curvature term, provided the hypersurface is mean convex.

15 pages; 3 figures