Discrete Ricci curvature: Open problems

This document lists some open problems related to the notion of discrete Ricci curvature defined in [Oll09, Oll07]. Do not hesitate to contact me for precisions. Please inform me if you seriously work on one of these problems, so that I don’t put a student on it! The problems are not ordered. Problem A (Log-concave measures). Ricci curvature is positive for RN equipped with a Gaussian measure, and this generalizes to smooth, uniformly strictly log-concave measures. What happens for a general log-concave measure? The next example would be a convex set (whose boundary has “positive curvature” in an intuitive geometric sense), with associated process a Brownian motion conditioned not to leave the convex body. Problem B (Finsler manifolds). Ricci curvature is 0 for RN equipped with an Lp norm. Does this give anything interesting in Finsler manifolds? (Compare [Oht] and forthcoming work by Ohta and Sturm using the displacement convexity definition.) Problem C (Nilpotent groups). Ricci curvature of ZN is 0. What happens on discrete or continuous nilpotent groups? For example, on the discrete Heisenberg group 〈 a, b, c | ac = ca, bc = cb, [a, b] = c 〉, the natural discrete random walk analogous to the hypoelliptic diffusion operator on the continuous Heisenberg group is the random walk generated by a and b. Since these generators are free up to length 8, clearly Ricci curvature is negative at small scales, but does it tend to 0 at larger and larger scales? Problem D (Continuous-time). In lots of examples, the natural process is a continuous-time one. When the space is finite or compact, or when one has good explicit knowledge of the process (as for Ornstein– Uhlenbeck on RN ), discretization works very well, but this might not be the case in full generality. Suppose a continuous-time Markov semigroup (mx)x∈X,t∈R+ is given. One can define a Ricci curvature in a straightforward manner as κ(x, y) := lim inf t→0+ 1 t d(x, y)− T1(mx,my) d(x, y)