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

in Bulletin des Sciences Mathématiques (2016), 140(5), 541-561

Let Lt:=Δt+ZtLt:=Δt+Zt for a C1,1C1,1-vector field Z on a differential manifold M possibly with a boundary ∂M , where ΔtΔt is the Laplacian operator induced by a time dependent metric gtgt differentiable ... [more ▼]

Let Lt:=Δt+ZtLt:=Δt+Zt for a C1,1C1,1-vector field Z on a differential manifold M possibly with a boundary ∂M , where ΔtΔt is the Laplacian operator induced by a time dependent metric gtgt differentiable in t∈[0,Tc)t∈[0,Tc). In this article, by constructing suitable coupling, transportation-cost inequalities on the path space of the (reflecting if ∂M≠∅∂M≠∅) diffusion process generated by LtLt are proved to be equivalent to a new curvature lower bound condition and the convexity of the geometric flow (i.e., the boundary keeps convex). Some of them are further extended to non-convex flows by using conformal changes of the flows. As an application, these results are applied to the Ricci flow with the umbilic boundary. [less ▲]

A stochastic approach to a priori estimates and Liouville theorems for harmonic maps

in Bulletin des Sciences Mathématiques (2011), 135(6-7), 816-843

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type ... [more ▼]

Nonlinear versions of Bismut type formulas for the differential of a harmonic map between Riemannian manifolds are used to establish a priori estimates for harmonic maps. A variety of Liouville type theorems is shown to follow as corollaries from such estimates by exhausting the domain through an increasing sequence of geodesic balls. This probabilistic method is well suited for proving sharp estimates under various curvature conditions. We discuss Liouville theorems for harmonic maps under the following conditions: small image, sublinear growth, non-positively curved targets, generalized bounded dilatation, Liouville manifolds as domains, certain asymptotic behaviour. [less ▲]