Noncoercive Ricci flow invariant curvature cones

TL;DR

This paper investigates specific noncoercive curvature conditions preserved under Ricci flow, revealing limitations on their strength and implications for Einstein and conformally flat manifolds in higher dimensions.

Contribution

It characterizes noncoercive Ricci flow invariant curvature conditions, showing their constraints and relation to scalar curvature and Einstein spaces in dimensions greater than 4.

Findings

Nonnegative scalar curvature is the only noncoercive Ricci flow invariant condition weaker than Einstein with nonnegative scalar curvature.

Stronger Ricci flow invariant conditions than nonnegative scalar curvature cannot hold on certain compact Einstein symmetric spaces.

Conditions satisfied by all conformally flat manifolds with nonnegative scalar curvature are characterized.

Abstract

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