Michael-Simon inequalities for $k$-th mean curvatures

TL;DR

This paper extends Alexandrov-Fenchel inequalities to $k$-curvature operators, employing optimal transport to establish Michael-Simon type inequalities for $k$-convex domains, advancing geometric analysis techniques.

Contribution

It introduces new Michael-Simon inequalities for $k$-curvature operators using optimal transport, expanding the understanding of curvature inequalities in convex geometry.

Findings

Established Michael-Simon inequalities for $k$-curvature operators.

Connected curvature inequalities with optimal transport methods.

Extended Alexandrov-Fenchel inequalities to $k$-convex domains.

Abstract

This paper continues the study of Alexandrov-Fenchel inequalities for quermassintegrals for $k$-convex domains. It focuses on the application to the Michael-Simon type inequalities for $k$-curvature operators. The proof uses optimal transport maps as a tool to relate curvature quantities defined on the boundary of a domain.