Nonexistence of NNSC fill-ins with large mean curvature

TL;DR

This paper proves that certain closed Riemannian manifolds cannot be filled with nonnegative scalar curvature if the mean curvature is sufficiently large at each point, extending to negative scalar curvature bounds.

Contribution

It establishes nonexistence results for fill-ins with prescribed scalar curvature and large mean curvature, advancing understanding of geometric constraints on fill-ins.

Findings

No fill-in with nonnegative scalar curvature exists for large point-wise mean curvature.

Similar nonexistence results hold for fill-ins with negative scalar curvature bounds.

Results contribute to the theory of scalar curvature and geometric fill-in problems.

Abstract

In this note we show that a closed Riemannian manifold does not admit a fill-in with nonnegative scalar curvature if the mean curvature is point-wise large. Similar result also holds for fill-ins with a negative scalar curvature lower bound.