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Abstract
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For a proper action by a locally compact group
on a manifold
with a
-equivariant
-structure,
we obtain obstructions to the existence of complete
-invariant
Riemannian metrics with uniformly positive scalar curvature. We focus on the case
where
is noncompact. The obstructions follow from a Callias-type index
theorem, and relate to positive scalar curvature near hypersurfaces in
.
We also deduce some other applications of this index theorem. If
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
-invariant
Riemannian metrics with uniformly positive scalar curvature, under an equivariant
bounded geometry assumption.
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Keywords
Callias operator, index, positive scalar curvature, proper
group action
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Mathematical Subject Classification 2010
Primary: 19K56
Secondary: 46L80, 53C27
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Milestones
Received: 19 February 2020
Revised: 11 November 2020
Accepted: 3 December 2020
Published: 1 August 2021
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