We show that the first betti number b1 (0) = d im H 1(0, !R) of a compact Riemannian orbifold 0 with Ricci curvature Ric(O ) ~ -(n 1)k and d iameter diam(O) :5 D is bounded above by a constant r (n, kD2 ) ~ 0 , depending only on dimension , curvature and diameter. In the case when t he orbi fold has nonnegative Ricci curvature, we show that the b1 (0) is bounded above by the dimension dim 0 , a...

In this paper, we prove that any non-flat ancient solution to KählerRicci flow with bounded nonnegative bisectional curvature has asymptotic volume ratio zero. We also classify all complete gradient shrinking solitons with nonnegative bisectional curvature. Both results generalize the corresponding earlier results of Perelman in [P1] and [P2]. The results then are applied to study the geometry ...

Since Milnor’s discovery of exotic spheres [Mi], one of the most intriguing problems in Riemannian geometry has been whether there are exotic spheres with positive curvature. It is well known that there are exotic spheres that do not even admit metrics with positive scalar curvature [Hi] . On the other hand, there are many examples of exotic spheres with positive Ricci curvature (cf. [Ch1], [He...

In this paper we introduce the concept of (ε)-almost paracontact manifolds, and in particular, of (ε)-para Sasakian manifolds. Several examples are presented. Some typical identities for curvature tensor and Ricci tensor of (ε)-para Sasakian manifolds are obtained. We prove that if a semi-Riemannian manifold is one of flat, proper recurrent or proper Ricci-recurrent, then it can not admit an (ε...

In this article we study the metric property and the function theory of asymptotically locally Euclidean (ALE) Kähler manifolds. In particular, we prove the Ricci flatness under the assumption that the Ricci curvature of such manifolds is either nonnegative or nonpositive. The result provides a generalization of previous gap type theorems established by Greene and Wu, Mok, Siu and Yau, etc. It ...

which is sharp as indicated in the Euclidean case. However even if M contains a small compact region where the Ricci curvature is not nonnegative, estimate (1.1) becomes very much different from (1.2) when r is large, due to the presence of the √ k term. Whether estimate (1.2) is stable under perturbation has been an open question for some time, in light of the known stability results on weaker...

The subject began in 1975, when Yau [Y1] proved that there are no nonconstant, positive harmonic functions on a complete manifold with nonnegative Ricci curvature. A few years later, Cheng [C] pointed out that using a local version of Yau’s gradient estimate, developed in his joint work with Yau [CY], one can show that there are no nonconstant harmonic functions of sublinear growth on a manifol...

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, ...

This paper presents an improved Euclidean Ricci flow method for spherical parameterization. We subsequently invent a scale space processing built upon Ricci energy to extract robust surface features for accurate surface registration. Since our method is based on the proposed Euclidean Ricci flow, it inherits the properties of Ricci flow such as conformality, robustness and intrinsicalness, faci...