# Positive scalar curvature and an equivariant Callias-type index theorem for proper actions

Research paper by Hao Guo, Peter Hochs, Varghese Mathai

Indexed on: 22 Jan '20Published on: 20 Jan '20Published in: arXiv - Mathematics - Differential Geometry

#### Abstract

For a proper action by a locally compact group $G$ on a manifold $M$ with a $G$-equivariant Spin-structure, we obtain obstructions to the existence of complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature. We focus on the case where $M/G$ is noncompact. The obstructions follow from a Callias-type index theorem, and relate to positive scalar curvature near hypersurfaces in $M$. If $G$ is a connected Lie group, then these obstructions vanish under a mild assumption on the action. In that case, we generalise a construction by Lawson and Yau to obtain complete $G$-invariant Riemannian metrics with uniformly positive scalar curvature, under an equivariant bounded geometry assumption.