Vol. 6, No. 2, 2021

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

Hao Guo, Peter Hochs and Varghese Mathai

Vol. 6 (2021), No. 2, 319–356
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 MG 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.

Keywords
Callias operator, index, positive scalar curvature, proper group action
Mathematical Subject Classification 2010
Primary: 19K56
Secondary: 46L80, 53C27
Milestones
Received: 19 February 2020
Revised: 11 November 2020
Accepted: 3 December 2020
Published: 1 August 2021
Authors
Hao Guo
Department of Mathematics
Texas A&M University
College Station, TX
United States
Peter Hochs
Institute for Mathematics, Astrophysics and Particle Physics
Radboud University
Nijmegen
Netherlands
Varghese Mathai
School of Mathematical Sciences
University of Adelaide
Adelaide SA
Australia