The Michael-Simon-Sobolev inequality on manifolds for positive symmetric tensor fields

TL;DR

This paper extends the Michael-Simon-Sobolev inequality to positive symmetric tensor fields on manifolds with nonnegative curvature, using the ABP method, generalizing previous work by Brendle.

Contribution

It introduces a new inequality for tensor fields on manifolds, broadening the scope of geometric analysis techniques in Riemannian geometry.

Findings

Proves the inequality for symmetric positive definite tensor fields.

Uses the Alexandrov-Bakelman-Pucci (ABP) method in a novel geometric context.

Generalizes previous scalar inequalities to tensor fields.

Abstract

We prove the Michael-Simon-Sobolev inequality for smooth symmetric uniformly positive definite (0, 2)-tensor fields on compact submanifolds with or without boundary in Riemannian manifolds with nonnegative sectional curvature by the Alexandrov-Bakelman-Pucci (ABP) method. It should be a generalization of S. Brendle in [2].