Topological obstructions to nonnegative scalar curvature and mean convex boundary

TL;DR

This paper investigates topological barriers preventing certain manifolds with boundary from supporting metrics with non-negative scalar curvature and mean convex boundary, providing explicit examples of such manifolds.

Contribution

It constructs numerous manifolds with boundary that cannot admit metrics with non-negative scalar curvature and mean convex boundary, highlighting topological obstructions.

Findings

Certain manifolds like $(T^{n-2}\times \Sigma)\# N$ do not admit such metrics.

Manifolds like $(I\times T^{n-1})\#N$ cannot support positive scalar curvature with mean convex boundary.

The work identifies specific topological conditions obstructing these geometric structures.

Abstract

We study topological obstructions to the existence of a Riemannian metric on manifolds with boundary such that the scalar curvature is non-negative and the boundary is mean convex. We construct many compact manifolds with boundary which admit no Riemannian metric with non-negative scalar curvature and mean convex boundary. For example, we show that the manifold $(T^{n-2}\times \Sigma )\# N$, where $\Sigma$ is a compact, connected and orientable surface which is not a disk or a cylinder and $N$ is a closed $n$-dimensional manifold, does not admit a metric of non-negative scalar curvature and mean convex boundary, and the manifold $(I\times T^{n-1})\#N$, where $I=[a,b]$, does not admit a metric of positive scalar curvature and mean convex boundary.