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

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.

Callias operator, index, positive scalar curvature, proper group action
Mathematical Subject Classification 2010
Primary: 19K56
Secondary: 46L80, 53C27
Received: 19 February 2020
Revised: 11 November 2020
Accepted: 3 December 2020
Published: 1 August 2021
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
Varghese Mathai
School of Mathematical Sciences
University of Adelaide
Adelaide SA