Boundary conditions for scalar curvature

TL;DR

This paper develops new obstructions and deformation principles for positive scalar curvature metrics on spin manifolds with boundaries, using index theory and homotopy techniques.

Contribution

It introduces an index-theoretic obstruction for scalar curvature with boundary conditions and establishes a deformation principle affecting the topology of metric spaces.

Findings

Obstruction to positive scalar curvature metrics with mean convex boundaries.

Deformation principle induces weak homotopy equivalences of metric spaces.

Existence of manifolds with nontrivial higher homotopy groups of scalar curvature metric spaces.

Abstract

Based on the Atiyah-Patodi-Singer index formula, we construct an obstruction to positive scalar curvature metrics with mean convex boundaries on spin manifolds of infinite K-area. We also characterize the extremal case. Next we show a general deformation principle for boundary conditions of metrics with lower scalar curvature bounds. This implies that the relaxation of boundary conditions often induces weak homotopy equivalences of spaces of such metrics. This can be used to refine the smoothing of codimension-one singularites a la Miao and the deformation of boundary conditions a la Brendle-Marques-Neves, among others. Finally, we construct compact manifolds for which the spaces of positive scalar curvature metrics with mean convex boundaries have nontrivial higher homotopy groups.