On compact $3$-manifolds with nonnegative scalar curvature with a CMC boundary component

TL;DR

This paper derives inequalities for compact 3-manifolds with nonnegative scalar curvature using the Riemannian Penrose inequality and positive mass theorem, focusing on boundaries with constant mean curvature and minimal surfaces.

Contribution

It introduces a collar extension method inspired by Mantoulidis-Schoen to analyze boundary inequalities in such manifolds.

Findings

Inequalities relating boundary geometry and scalar curvature.

Construction of collar extensions for boundary analysis.

Application of Penrose inequality and positive mass theorem.

Abstract

We apply the Riemannian Penrose inequality and the Riemannian positive mass theorem to derive inequalities on the boundary of a class of compact Riemannian $3$-manifolds with nonnegative scalar curvature. The boundary of such a manifold has a CMC component, i.e. a $2$-sphere with positive constant mean curvature; and the rest of the boundary, if nonempty, consists of closed minimal surfaces. A key step in our proof is the construction of a collar extension that is inspired by the method of Mantoulidis-Schoen \cite{M-S}.