Scalar curvature deformations with non-compact boundaries

Abstract

We develop a general deformation principle for families of Riemannian metrics on smooth manifolds with possibly non-compact boundary, preserving lower scalar curvature bounds. The principle is used in order to strengthen boundary conditions, from mean convex to totally geodesic or doubling. The deformation principle preserves further geometric properties such as completeness and a given quasi-isometry type. As an application, we prove non-existence results for Riemannian metrics with (uniformly) positive scalar curvature and mean convex boundary, including progress on a generalised Geroch conjecture with boundary and some investigation of the Whitehead manifold.