The doubling conjecture for positive scalar curvature

TL;DR

This paper proves the doubling conjecture for positive scalar curvature under certain fundamental group conditions and explores metric adjustments for embedded hypersurfaces.

Contribution

It establishes the conjecture for manifolds with specific fundamental group split conditions and develops techniques for metric adjustments on hypersurfaces.

Findings

The doubling conjecture holds for manifolds with split-condition fundamental groups.

Surgery techniques can produce positive scalar curvature metrics with mean convex boundary.

Adjustments of psc-metrics can make hypersurfaces minimal or totally geodesic in many cases.

Abstract

The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the boundary satisfies a certain split-condition on fundamental groups. Our proof is based on surgery-techniques for positive scalar and mean curvature. If the boundary is non-connected, we use existence of area-minimizing hypersurfaces and the monotonicity-formula. Furthermore, we investigate if a psc-metric on a closed manifold can be adjusted so that a given embedded hypersurface is minimal, stable minimal or totally geodesic. While not true in general, such an adjustment is possible in many cases.