Reflecting Diffusion Processes on Manifolds Carrying Geometric Flow

Let (L_t:=Delta _t+Z_t) for a (C^{infty })-vector field Z on a differentiable manifold M with boundary (partial M), where (Delta _t) is the Laplacian operator, induced by a time dependent metric (g_t) differentiable in (tin [0,T_mathrm {c})). We first establish the derivative formula for the associated reflecting diffusion semigroup generated by (L_t). Then, by using parallel displacement and reflection, we construct the couplings for the reflecting (L_t)-diffusion processes, which are applied to gradient estimates and Harnack inequalities of the associated heat semigroup. Finally, as applications of the derivative formula, we present a number of equivalent inequalities for a new curvature lower bound and the convexity of the boundary. These inequalities include the gradient estimates, Harnack inequalities, transportation-cost inequalities and other functional inequalities for diffusion semigroups.