Sobolev inequalities involving 2-tensor fields in manifolds with nonnegative sectional curvature

TL;DR

This paper develops new Sobolev inequalities for tensor fields on manifolds with nonnegative sectional curvature using the ABP method, extending classical inequalities to tensor settings.

Contribution

It introduces Log Sobolev and Michael Simon Sobolev inequalities for tensor fields in curved manifolds, a novel extension of existing scalar inequalities.

Findings

Established Log Sobolev inequality for tensor fields.

Proved Michael Simon Sobolev inequality in the tensor context.

Applied the ABP method to tensor fields in geometric analysis.

Abstract

By applying the ABP method, we establish both Log Sobolev type inequality and Michael Simon Sobolev inequality for smooth symmetric uniformly positive definite (0,2) tensor fields in manifolds with nonnegative sectional curvature.