Interior control for surfaces with positive scalar curvature and its application

TL;DR

This paper proves that certain aspherical manifolds with specific submanifold configurations cannot support complete metrics with positive scalar curvature, introducing a new interior control technique for surfaces with positive scalar curvature.

Contribution

It introduces a novel interior control method for the extrinsic diameter of surfaces with positive scalar curvature and applies it to obstruct positive scalar curvature metrics on certain manifolds.

Findings

No positive scalar curvature metrics on 3- and 4-dimensional aspherical manifolds with specified submanifolds.

Extension of the obstruction to 5-dimensional cases with codimension 1 or 2 submanifolds.

Development of a new interior control technique for surfaces with positive scalar curvature.

Abstract

Let $M^{n}$, $n\in\{3,4,5\}$, be a closed aspherical $n$-manifold and $S\subset M$ a subset consisting of disjoint incompressible embedded closed aspherical submanifolds (possibly with different dimensions). When $n =3,4$, we show that $M\setminus S$ cannot admit any complete metric with positive scalar curvature. When $n=5$, we obtain the same result when $S$ contains a submanifold of codimension 1 or 2. The key ingredient is a new interior control for the extrinsic diameter of surfaces with positive scalar curvature.