Abstract. We consider the framework used by Bakry and Emery in their work on logarithmic Sobolev inequalities to define a notion of coarse Ricci curvature on smooth metric measure spaces alternative to the notion proposed by Y. Ollivier. We discuss applications of our construction to the manifold learning problem, specifically to the statistical problem of estimating the Ricci curvature of a su...

We give a new estimate on the lower bound for the first Dirichlet eigenvalue for the compact manifolds with boundary and positive Ricci curvature in terms of the diameter and the lower bound of the Ricci curvature and give an affirmative answer to the conjecture of P. Li for the Dirichlet eigenvalue.

We prove that for a solution (M, g(t)), t ∈ [0, T ), where T < ∞, to the Ricci flow on a complete non-compact Riemannian manifold with the Ricci curvature tensor uniformly bounded by some constant C on M × [0, T ), the curvature tensor stays uniformly bounded on M × [0, T ).

Consider a sample of n points taken i.i.d from a submanifold of Euclidean space. This defines a metric measure space. We show that there is an explicit set of scales tn → 0 such that a coarse Ricci curvature at scale tn on this metric measure space converges almost surely to the coarse Ricci curvature of the underlying manifold.

The main result of the paper is a computation of the Ricci curvature of Diff(S)/S. Unlike earlier results on the subject, we do not use the Kähler structure symmetries to compute the Ricci curvature, but rather rely on classical finite-dimensional results of Nomizu et al on Riemannian geometry of homogeneous spaces. Table of

Unique continuation results are proved for metrics with prescribed Ricci curvature in the setting of bounded metrics on compact manifolds with boundary, and in the setting of complete conformally compact metrics on such manifolds. In addition, it is shown that the Ricci curvature forms an elliptic system in geodesic-harmonic coordinates naturally associated with the boundary data.

We show that a four-dimensional complete gradient shrinking Ricci soliton with positive isotropic curvature is either a quotient of S4 or a quotient of S3 × R. This gives a clean classification result removing the earlier additional assumptions in [14] by Wallach and the second author. The proof also gives a classification result on gradient shrinking Ricci solitons with nonnegative isotropic c...

We estimate the lower bound of the first non-zero eigenvalue of a compact Riemannian manifold with negative lower bound of Ricci curvature in terms of the diameter and the lower bound of Ricci curvature and give an affirmative answer to the conjecture of H. C. Yang.

Along the line of the Yang Conjecture, we give a new estimate on the lower bound of the first non-zero eigenvalue of a closed Riemannian manifold with negative lower bound of Ricci curvature in terms of the in-diameter and the lower bound of Ricci curvature.

In this announcement, we describe some results of an ongoing investigation of function theory on spaces with a lower Ricci curvature bound. In particular, we announce results on harmonic functions of polynomial growth on open manifolds with nonnegative Ricci curvature and Euclidean volume growth.