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.Question 1.1.When does M admit a complete, G-invariant Riemannian metric with uniformly positive scalar curvature?We are mainly interested in the case where M/G is noncompact.

Locations

• Annals of K-Theory - View
• arXiv (Cornell University) - View - PDF
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