Diffusion semigroup on manifolds with time-dependent metrics

Let $L_t:=\Delta_t +Z_t $, $t\in [0,T_c)$ on a differential manifold equipped with a complete geometric flow $(g_t)_{t\in [0,T_c)}$, where $\Delta_t$ is the Laplacian operator induced by the metric $g_t$ and $(Z_t)_{t\in [0,T_c)}$ is a family of $C^{1,\infty}$-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 $L_t$. To this end, we establish the derivative formula for the associated semigroup and construct couplings for these diffusion processes by parallel displacement and reflection.