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Type: Journal article
Title: Positive scalar curvature and an equivariant Callias-type index theorem for proper actions
Author: Guo, H.
Hochs, P.
Varghese, M.
Citation: Annals of K-Theory, 2021; 6(2):319-356
Publisher: Mathematical Sciences Publishers
Issue Date: 2021
ISSN: 2379-1683
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: 10.2140/akt.2021.6.319
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