Non-coercive Ricci flow invariant curvature cones
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- by Thomas Richard and Harish Seshadri PDF
- Proc. Amer. Math. Soc. 143 (2015), 2661-2674 Request permission
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
- MR Author ID: 712201
- Email: harish@math.iisc.ernet.in
- 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
- © Copyright 2015 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 143 (2015), 2661-2674
- MSC (2010): Primary 53C44
- DOI: https://doi.org/10.1090/S0002-9939-2015-12619-6
- MathSciNet review: 3326045