Curvature cones and the Ricci flow.

Abstract

This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points.First we describe the known examples of preserved curvature conditions and how they have been used to derive geometric results, in particular sphere theorems.We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters.The Ricci flow is the following evolution equation:where (g(t)) t∈[0,T ) is a one-parameter family of smooth Riemannian metrics on a fixed manifold M , and g 0 is a given smooth Riemannian metric on M . It was introduced by R. Hamilton in 1982 ([Ham82]), where it was used to study the topology of compact 3-manifolds with positive Ricci curvature. Analytically, the Ricci flow is a degenerate parabolic system. Existence and uniqueness for the Cauchy problem (1) . Since Hamilton's work, the Ricci flow has been used to solve various geometric problems. We refer to the previously cited books for examples. Here we will just briefly mention two of the biggest geometric achievements of Ricci flow:• The proof of Thurston's Geometrization conjecture for 3-manifolds by G. The objects we will be dealing with in this survey have little to do with Perelman's work, but were pivotal in the proof of the differentiable sphere theorem.A priori estimates are among the most basic tools in the study of PDEs, one can divide them into two classes: integral estimates and pointwise estimates. Pointwise estimates often come from suitable maximum principles and are thus more often encountered in the realm of parabolic or elliptic equations. The Ricci flow being a geometric parabolic PDE, it is tempting to look for geometrically meaningful pointwise estimates for Ricci flows.If one looks at Hamilton's foundational work, one sees that after having proven short time existence and derived some variation formulas for the Ricci flow, Hamilton proves the following result:Proposition 0.1. Let (M 3 , g 0 ) be a compact 3-manifold with nonnegative Ricci curvature, then the solution g(t) to (1) with initial condition g 0 satisfies Ric g(t) ≥ 0 for all t ≥ 0. This is the kind of geometric pointwise estimate we will be concerned with. More precisely, we will try to gather what is known about various nonnegativity properties of the curvature of Riemannian manifold (M, g 0 ) which remain valid for solutions g(t) of the Ricci flow starting at g 0 .Let us now describe the contents of this paper. Section 1 sets the scene by introducing an abstract framework which allows us to consider nonnegativity conditions on the curvature as convex cones inside some vector space satisfying some invariance properties: the so-called curvature cones. We then describe how Hamilton's maximum principle characterizes which curvature cones lead to a nonnegativity condition on the curvature which is preserved under the action of the Ricci flow. In section 2 we review the...

The commutative nonassociative algebra of metric curvature tensors

“…When comparing with the literature, care has to be taken. For example, that some signs in [32] at first glance appear inconsistent with those here is because [32] works directly with curvature operators, essentially with what is here called X ∧ 2 V * , which is the image under the map here called ♯ of the tensor − 1 2 X 𝑖 𝑗𝑘𝑙 , whereas here results are stated directly in terms of the tensor X 𝑖 𝑗𝑘𝑙 .…”

“…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.…”