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Type: Journal article
Title: Positive scalar curvature and callias-type index theorems for proper actions
Author: Guo, H.
Hochs, P.
Varghese, M.
Citation: Bulletin of the Australian Mathematical Society, 2019; 99(2):342-343
Publisher: Cambridge University Press (CUP)
Issue Date: 2019
ISSN: 0004-9727
Statement of
Hao Guo, Peter Hochs and Varghese Mathai
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.
Keywords: Callias operator; index; positive scalar curvature; proper group action
Rights: © Copyright 2021 Mathematical Sciences Publishers. All rights reserved.
DOI: 10.1017/S0004972718001338
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