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Non-coercive Ricci flow invariant curvature cones


Authors: Thomas Richard and Harish Seshadri
Journal: Proc. Amer. Math. Soc. 143 (2015), 2661-2674
MSC (2010): Primary 53C44
DOI: https://doi.org/10.1090/S0002-9939-2015-12619-6
Published electronically: February 16, 2015
MathSciNet review: 3326045
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Abstract: This note is a study of nonnegativity conditions on curvature preserved by the Ricci flow. We focus on a specific class of curvature conditions which we call non-coercive: These are the conditions for which nonnegative curvature and vanishing scalar curvature does not imply flatness.

We show, in dimensions greater than $ 4$, that if a Ricci flow invariant nonnegativity condition is satisfied by all Einstein curvature operators with nonnegative scalar curvature, then this condition is just the nonnegativity of scalar curvature. As a corollary, we obtain that a Ricci flow invariant curvature condition, which is stronger than a nonnegative scalar curvature, cannot be strictly satisfied by curvature operators (other than multiples of the identity) of compact Einstein symmetric spaces.

We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.


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Additional Information

Thomas Richard
Affiliation: Department of Mathematics, Indian Institute of Science, 560012 Bangalore, India
Email: thomas@math.iisc.ernet.in, thomas.richard@u-pec.fr

Harish Seshadri
Affiliation: Department of Mathematics, Indian Institute of Science, 560012 Bangalore, India
Email: harish@math.iisc.ernet.in

DOI: https://doi.org/10.1090/S0002-9939-2015-12619-6
Received by editor(s): August 15, 2013
Received by editor(s) in revised form: January 8, 2014
Published electronically: February 16, 2015
Communicated by: Lei Ni
Article copyright: © Copyright 2015 American Mathematical Society