Sobolev Inequalities In Manifolds With Asymptotically Nonnegative Curvature

TL;DR

This paper establishes sharp Sobolev and isoperimetric inequalities for submanifolds in manifolds with asymptotically nonnegative curvature, extending previous results to a broader class of manifolds.

Contribution

It generalizes Brendle's inequalities to manifolds with asymptotically nonnegative curvature using the ABP-method.

Findings

Sharp Sobolev inequalities for asymptotically nonnegative curvature manifolds

Generalization of Brendle's inequalities

Applications to geometric analysis

Abstract

Using the ABP-method as in a recent work by Brendle, we establish some sharp Sobolev and isoperimetric inequalities for compact domains and submanifolds in a complete Riemannian manifold with asymptotically nonnegative curvature. These inequalities generalize those given by Brendle in the case of complete Riemannian manifolds with nonnegative curvature.