Heat Flow and Concentration of Measure on Directed Graphs with a Lower Ricci Curvature Bound

Measure Rigidity of Ricci Curvature Lower Bounds

The measure contraction property, MCP for short, is a weak Ricci curvature lower bound conditions for metric measure spaces. The goal of this paper is to understand which structural properties such assumption (or even weaker modifications) implies on the measure, on its support and on the geodesics of the space. We start our investigation from the euclidean case by proving that if a positive Ra...

Ricci Lower Bound for Kähler–ricci Flow

Heat flow and calculus on metric measure spaces with Ricci curvature bounded below - the compact case

We provide a quick overview of various calculus tools and of the main results concerning the heat flow on compact metric measure spaces, with applications to spaces with lower Ricci curvature bounds. Topics include the Hopf-Lax semigroup and the Hamilton-Jacobi equation in metric spaces, a new approach to differentiation and to the theory of Sobolev spaces over metric measure spaces, the equiva...

Local Curvature Bound in Ricci Flow

Theorem 2 Given n and v0 > 0, there exists ǫ0 > 0 depending only on n and v0 which has the following property. For any r0 > 0 and ǫ ∈ (0, ǫ0] suppose (Mn, g(t)) is a complete smooth solution to the Ricci flow on [0, (ǫr0) 2] with bounded sectional curvature, and assume that at t = 0 for some x0 ∈ M we have curvature bound |Rm |(x, 0) ≤ r 0 for all x ∈ Bg(0)(x0, r0), and volume lower bound Volg(...

A Lower Bound for the Scalar Curvature of the Standard Solution of the Ricci Flow

In this paper we will give a rigorous proof of the lower bound for the scalar curvature of the standard solution of the Ricci flow conjectured by G. Perelman. We will prove that the scalar curvature R of the standard solution satisfies R(x, t) ≥ C0/(1−t) ∀x ∈ R , 0 ≤ t < 1, for some constant C0 > 0. Recently there is a lot of study of Ricci flow on manifolds by R. Hamilton [H1-6], S.Y. Hsu [Hs1...