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

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 $M$. We also deduce some other applications of this index theorem. If $G$ is a connected Lie group, then the obstructions to positive scalar curvature 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.

arXiv comment36 pages, 5 figures. Some applications added in v2

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Cited by in corpus (2)

• Functoriality for higher rho invariants of elliptic operators
• Higher localised $\hat{A}$-genera for proper actions and applications