Li Juan Cheng

A probabilistic method for gradient estimates of some geometric flows

In general, gradient estimates are very important and necessary for deriving convergence results in different geometric flows, and most of them are obtained by analytic methods. In this paper, we will apply a stochastic approach to systematically give gradient estimates for some important geometric quantities under the Ricci flow, the mean curvature flow, the forced mean curvature flow and the Yamabe flow respectively. Our conclusion gives another example that probabilistic tools can be used to simplify proofs for some problems in geometric analysis.

DIFFUSION SEMIGROUP ON MANIFOLDS WITH TIME-DEPENDENT METRICS

Abstract Let L t := Δ t + Z t {L_{t}:=\Delta_{t}+Z_{t}} , t ∈ [ 0 , T c ) {t\in[0,T_{c})} on a differential manifold equipped with a complete geometric flow ( g t ) t ∈ [ 0 , T c ) {(g_{t})_{t\in[0,T_{c})}} , where Δ t {\Delta_{t}} is the Laplacian operator induced by the metric g t {g_{t}} and ( Z t ) t ∈ [ 0 , T c ) {(Z_{t})_{t\in[0,T_{c})}} is a family of C 1 , ∞ {C^{1,\infty}} -vector fields. In this article, we present a number of equivalent inequalities for the lower bound curvature condition, which include gradient inequalities, transportation-cost inequalities, Harnack inequalities and other functional inequalities for the semigroup associated with diffusion processes generated by L t {L_{t}} . To this end, we establish derivative formulae for the associated semigroup and construct coupling processes for these diffusion processes by parallel displacement and reflection.

Evolution systems of measures and semigroup properties on evolving manifolds

An evolving Riemannian manifold

Characterization of Pinched Ricci Curvature by Functional Inequalities

(M,g_t)_{t\in I}

Functional inequalities on manifolds with non-convex boundary

consists of a smooth

Eigentime identity for one-dimensional diffusion processes

d

Weak poincaré inequality for convolution probability measures

-dimensional manifold

Transportation-cost inequalities on path spaces over manifolds carrying geometric flows

M

An integration by parts formula on path space over manifolds carrying geometric flow

, equipped with a geometric flow

The Radial Part of Brownian Motion with Respect to \mathcal L -Distance Under Ricci Flow

g_t