  Let $L_t:=\Delta_t+Z_t$ for a $C^{1,1}$-vector field $Z$ on a differential
manifold $M$ with boundary $\partial M$, where $\Delta_t$ is the Laplacian
induced by a time dependent metric $g_t$ differentiable in $t\in [0,T_c)$. We
first introduce the reflecting diffusion process generated by $L_t$ and
establish the derivative formula for the associated diffusion semigroup; then
construct the couplings for the reflecting $L_t$-diffusion processes by
parallel and reflecting displacement, which implies the gradient estimates of
the associated heat semigroup; and finally, present a number of equivalent
inequalities for the curvature lower bound and the convexity of the boundary,
including the gradient estimations, Harnack inequalities, transportation-cost
inequalities and other functional inequalities for diffusion semigroup.
