Non-coercive Ricci flow invariant curvature cones

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 (if not void) the condition "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 2 × S 2 or CP 2 . We also investigate conditions which are satisfied by all conformally flat manifolds with nonnegative scalar curvature.When studying Ricci flow, it is useful to know that some "nonnegative curvature"-type geometric condition is preserved along the flow. Although Ricci flow has been studied extensively since R. Hamilton's seminal paper, there is still no comprehensive theory of curvature conditions which are preserved by Ricci flow. A significant advance in this direction is the work of Wilking ([Wil10]) which gives a unified construction for almost all known Ricci flow invariant curvature conditions. The paper [GMS11] gives general results on curvature conditions coming from this construction.We want to gain a better understanding of general Ricci flow invariant curvature conditions. Curvature conditions are encoded by convex cones C (called curvature cones) in the space of algebraic curvature operators S 2 B Λ 2 R n which are invariant under the natural action of the orthogonal group. As a consequence of the maximum principle for systems, a sufficient condition for a curvature condition to be preserved under the Ricci flow is the preservation of the cone C by the flow of some explicit vector field. Readers not familiar with these notions will find a quick exposition and references in section 1.

Positive isotropic curvature and self-duality in dimension 4

“…Arguing as in the proof of corollary 0.3 in [22], we get that W + ⊂ C. Similarly, using that C contains some R which is not in N N IC − , one can show that the curvature operator of CP 2 is in the interior of C and get that W − ⊂ C. We have thus proved that W ⊂ C. This proves the first part of the theorem.…”

“…W − ). Proposition 3.3 (Proposition 3.6 from [22]). If an oriented curvature cone C contains W and is Ricci flow invariant, then C is the cone C Scal of curvature operators whose scalar curvature is nonnegative.…”

“…We will need a couple of results from [22]. Proposition 3.2 (Proposition A.5 from [22]). If an oriented curvature cone C contains a non zero R ∈ W + (resp.…”

The commutative nonassociative algebra of metric curvature tensors

“…For background on the definition of * as in (5.14), its properties, and its role in the study of the Ricci flow, see also [20]. Some features of the algebra (MC(V * ), * ) are used implicitly in the study of the Ricci flow [2,4,5,17,18,21,32,33,34,41]. The algebraic perspective makes some of the manipulations used in such studies appear more natural and focuses attention on certain structural 1Due to a typographical error, its Theorem 2 is mislabeled as Theorem 3.…”

“…Remark 1.2. Some authors [2,32,34] define * directly in terms of curvature operators on 2 V * . Here * is defined on curvature tensors, and the two definitions involve curvature operators on 𝑆 2 V * and MC(V * ) itself.…”