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2013

Noncoercive Ricci flow invariant curvature cones

Richard, Thomas, Seshadri, Harish

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This note is a study of nonnegativity conditions on curvature which are preserved by the Ricci flow.

We focus on specific kinds of curvature conditions which we call noncoercive, these are the conditions for which nonnegative curvature and vanishing scalar curvature doesn't imply flatness.
We show that, in dimensions greater than 4, if a Ricci flow invariant condition is weaker than "Einstein with nonnegative scalar curvature", then this condition has to be "nonnegative scalar curvature".
As a corollary, we obtain that a Ricci flow invariant curvature condition which is stronger than "nonnegative scalar curvature" cannot be (strictly) satisfied by compact Einstein symmetric spaces such as S^2xS^2 or CP^2.

Noncoercive Ricci flow invariant curvature cones · Around

Built on

Four-manifolds with positive curvature operator
Richard S. Hamilton · 1986
Earlier work this paper cites.
Minimal two-spheres and the topology of manifolds with positive curvature on totally isotropic two-planes
Mario J. Micallef and John Douglas Moore · 1988
Earlier work this paper cites.
Sign and geometric meaning of curvature
M. Gromov · 1991
Earlier work this paper cites.
Manifolds with positive curvature operators are space forms
Christoph Böhm and Burkhard Wilking · 2008
Earlier work this paper cites.

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Then

A Lie algebraic approach to Ricci flow invariant curvature conditions and Harnack inequalities
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Einstein metrics and preserved curvature conditions for the ricci flow
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