A note on scalar curvature and the convexity of boundaries

TL;DR

This paper demonstrates how to extend smooth Riemannian manifolds with non-negative scalar curvature beyond their boundary to achieve various convexity conditions, aiding in positive mass theorems.

Contribution

It introduces a method to extend manifolds with non-negative scalar curvature to alter boundary convexity properties while preserving curvature conditions.

Findings

Extensions can produce strictly convex, totally geodesic, or strictly concave boundaries.

The method applies to positive mass theorems.

Extensions preserve non-negative scalar curvature.

Abstract

We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geodesic or strictly concave. The extension procedure can be applied for instance to "positive mass" type of theorems.