Scalar-mean rigidity theorem for compact manifolds with boundary

TL;DR

This paper establishes a scalar-mean rigidity theorem for compact manifolds with boundary in dimensions less than five, extending previous scalar curvature methods and deriving sharp geometric rigidity results.

Contribution

It develops a dimension reduction technique for mean curvature, extending Schoen-Yau's scalar curvature approach to scalar-mean rigidity in low dimensions.

Findings

Proves scalar-mean rigidity theorem for manifolds with boundary in <5 dimensions.

Establishes sharp spherical radius rigidity and NNSC fill-in results.

Introduces a Lipschitz scalar-mean rigidity theorem.

Abstract

We prove a scalar-mean rigidity theorem for compact Riemannian manifolds with boundary in dimension less than five by developing a dimension reduction argument for mean curvature, which extends Schoen-Yau's dimension reduction argument for scalar curvature. As a corollary, we prove the sharp spherical radius rigidity theorem and best NNSC fill-in in terms of the mean curvature. Moreover, we prove a Lipschitz Listing type scalar-mean rigidity theorem for these dimensions.