Interior control for surfaces with positive scalar curvature and its application

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.

arXiv commentv3: Final version. 19 pages, 1 figure. To appear in Comm. Anal. Geom