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By using a coupling method, an explicit log-Harnack inequality with localgeometry quantities is established for (sub-Markovian) diffusion semigroups ona Riemannian manifold (possibly with boundary). This inequality as well as theconsequent $L^2$-gradient inequality, are proved to be equivalent to thepointwise curvature lower bound condition together with the convexity orabsence of the boundary. Some applications of the log-Harnack inequality arealso introduced.

Equivalent Harnack and Gradient Inequalities for Pointwise Curvature Lower Bound