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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 ofcomplete $G$-invariant Riemannian metrics with uniformly positive scalarcurvature. We focus on the case where $M/G$ is noncompact. The obstructionsfollow from a Callias-type index theorem, and relate to positive scalarcurvature near hypersurfaces in $M$. We also deduce some other applications ofthis index theorem. If $G$ is a connected Lie group, then the obstructions topositive scalar curvature vanish under a mild assumption on the action. In thatcase, 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.

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