ABSTRACT. In this article, functional inequalities for diffusion semigroups on Riemannian manifolds (possibly with boundary) are established, which are equivalent to pinched Ricci curvature, along with gradient estimates, Lp-inequalities and log-Sobolev inequalities. These results are further extended to differential manifolds carrying geometric flows. As application, it is shown that they can ...

In earlier work [2], we derived formal matched asymptotic profiles for families of Ricci flow solutions developing Type-II degenerate neckpinches. In the present work, we prove that there do exist Ricci flow solutions that develop singularities modeled on each such profile. In particular, we show that for each positive integer k ≥ 3, there exist compact solutions in all dimensions m ≥ 3 that be...

Consider a sequence of pointed n–dimensional complete Riemannian manifolds {(Mi, gi(t), Oi)} such that t ∈ [0, T ] are solutions to the Ricci flow and gi(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n–dimensional solution to the Ricci...

Ricci flow deformation of cosmological initial data sets in general relativity is a technique for generating families of initial data sets which potentially would allow to interpolate between distinct spacetimes. This idea has been around since the appearance of the Ricci flow on the scene, but it has been difficult to turn it into a sound mathematical procedure. In this expository talk we illu...

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 w...

Spectrum of kinematic fast dynamo operators in Ricci compressible flows in Einstein 2-manifolds is investigated. A similar expression, to the one obtained by Chicone, Latushkin and Montgomery-Smith (Comm Math Phys (1995)) is given, for the fast dynamo operator. The operator eigenvalue is obtained in a highly conducting media, in terms of linear and nonlinear orders of Ricci scalar. Eigenvalue s...

§0 Introduction. In a recent paper of Perelman[P1], an entropy formula for Ricci flow was derived. The formula turns out being of fundamental importance in the study of Ricci flow (cf. [P1, Sections 3, 4, 10]) as well as the Kähler-Ricci flow [P2]. The derivation of the entropy formula in [P1, Section 9] resembles the gradient estimate for the linear heat equation proved by Li-Yau in another fu...

Motivated by M\"uller-Haslhofer results on the dynamical stability and instability of Ricci-flat metrics under Ricci flow, we obtain for pairs vanishing 3-forms generalized flow.

These notes provides some details on the lectures 2,3,4 on the Ricci flow with surgery. They are not complete and probably contains some inaccuracies. Interested readers can find most exhaustives explanations on the Perelman's papers in [KL]. The aim of these lecture is to give the classification and the description of 3-dimensional κ-solutions. Let κ > 0 and (M n , g(t)) a solution of the Ricc...

Consider the Kähler-Ricci flow with finite time singularities over any closed Kähler manifold. We prove the existence of the flow limit in the sense of current towards the time of singularity. This answers affirmatively a problem raised by Tian in [23] on the uniqueness of the weak limit from sequential convergence construction. The notion of minimal singularity introduced by Demailly in the st...