Positive scalar curvature on manifolds with boundary and their doubles

TL;DR

This paper investigates conditions under which compact manifolds with boundary admit positive scalar curvature metrics, exploring boundary conditions and their influence on the scalar curvature of the doubled manifold.

Contribution

It provides complete characterizations of when such manifolds admit positive scalar curvature metrics with specific boundary conditions, linking boundary geometry to the scalar curvature of the double.

Findings

Characterizes when manifolds with boundary admit positive scalar curvature metrics with product structure near boundary.

Establishes conditions for positive scalar curvature with positive mean curvature boundary metrics.

Analyzes the relationship between boundary conditions and the scalar curvature of the doubled manifold.

Abstract

This paper is about positive scalar curvature on a compact manifold $X$ with non-empty boundary $\partial X$. In some cases, we completely answer the question of when $X$ has a positive scalar curvature metric which is a product metric near $\partial X$, or when $X$ has a positive scalar curvature metric with positive mean curvature on the boundary, and more generally, we study the relationship between boundary conditions on $\partial X$ for positive scalar curvature metrics on $X$ and the positive scalar curvature problem for the double $M=\operatorname{Dbl}(X,\partial X)$.