In this paper, we first derive the CR analogue of matrix Li–Yau–Hamilton inequality for the positive solution to the CR heat equation in a closed pseudohermitian (2n + 1)-manifold with nonnegative bisectional curvature and bitorsional tensor. We then obtain the CR Li–Yau gradient estimate in the Heisenberg group. We apply this CR gradient estimate and extend the CR matrix Li–Yau–Hamilton inequality to the case of the Heisenberg group. As a consequence, we derive the Hessian comparison property for the Heisenberg group.

We define the notion of a Ricci curvature lower bound for parametrized statistical models. Following the seminal ideas of Lott–Sturm–Villani, we define this notion based on the geodesic convexity of the Kullback–Leibler divergence in a Wasserstein statistical manifold, that is, a manifold of probability distributions endowed with a Wasserstein metric tensor structure. Within these definitions, which are based on Fisher information matrix and Wasserstein Christoffel symbols, the Ricci curvature is related to both, information geometry and Wasserstein geometry. These definitions allow us to formulate bounds on the convergence rate of Wasserstein gradient flows and information functional inequalities in parameter space. We discuss examples of Ricci curvature lower bounds and convergence rates in exponential family models.

We consider an ancient solution g(·, t) of the Ricci flow on a compact surface that exists for t 2 (−1, T ) and becomes spherical at time t = T . We prove that the metric g(·, t) is either a family of contracting spheres, which is a type I ancient solution, or a King–Rosenau solution, which is a type II ancient solution.

In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M × R2 has nonnegative isotropic curvature.

Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants-k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justi-fication of such a statement.