Ricci curvature,radial curvature and large volume growth

In this paper, we study complete noncompact Riemannian manifolds with Ricci curvature bounded from below. When the Ricci curvature is nonnegative, we show that this kind of manifolds are diffeomorphic to a Euclidean space, by assuming an upper bound on the radial curvature and a volume growth condition of their geodesic balls. When the Ricci curvature only has a lower bound, we also prove that such a manifold is diffeomorphic to a Euclidean space if the radial curvature is bounded from below. Moreover, by assuming different conditions and applying different methods, we shall prove more results on Riemannian manifolds with large volume growth.     

Rigidity and sphere theorem for manifolds with positive Ricci curvature

LetM be a complete Riemannian manifold with Ricci curvature having a positive lower bound. In this paper, we prove some rigidity theorems forM by the existence of a nice minimal hypersurface and a sphere theorem aboutM. We also generalize a Myers theorem stating that there is no closed immersed minimal submanifolds in an open hemisphere to the case that the ambient space is a complete Riemannian manifold withk-th Ricci curvature having a positive lower bound. Supported by the JSPS postdoctoral fellowship and NSF of China     

Rigidity of compact manifolds with boundary and nonnegative Ricci curvature

Let be an ()-dimensional compact Riemannian manifold with nonnegative Ricci curvature and nonempty boundary . Assume that the principal curvatures of are bounded from below by a positive constant . In this paper, we prove that the first nonzero eigenvalue of the Laplacian of acting on functions on satisfies with equality holding if and only if is isometric to an -dimensional Euclidean ball of radius . Some related rigidity theorems for are also proved.

     

The manifolds with nonnegative Ricci curvature and collapsing volume

Let be a complete noncompact -manifold with collapsing volume and . The paper proves that is of finite topological type under some restrictions on volume growth.

     

在$L^p$条件下完备Ricci孤立子流形的刚性定理

我们证明了在一定曲率和$L^p$条件下完备Ricci孤立子流形的一些刚性结果.     

Open manifolds with nonnegative Ricci curvature and large volume growth

In this paper, we study complete open n-dimensional Riemannian manifolds with nonnegative Ricci curvature and large volume growth. We prove among other things that such a manifold is diffeomorphic to a Euclidean n-space if its sectional curvature is bounded from below and the volume growth of geodesic balls around some point is not too far from that of the balls in . Received: August 17, 1998.     

Volume entropy based on integral Ricci curvature and volume ratio

For a compact Riemannian manifold M, we obtain an explicit upper bound of the volume entropy with an integral of Ricci curvature on M and a volume ratio between two balls in the universal covering space.     

Constraints on singularities under Ricci curvature bounds

We announce results giving constraints on the singularities of spaces which are Gromov-Hausdorf'f limits of sequences of Riemannian manifolds whose Ricci curvature and volume are bounded from below and whose curvature tensor is bounded in an integral sense.     

Ricci curvature and measures

In the last thirty years three a priori very different fields of mathematics, optimal transport theory, Riemannian geometry and probability theory, have come together in a remarkable way, leading to a very substantial improvement of our understanding of what may look like a very specific question, namely the analysis of spaces whose Ricci curvature admits a lower bound. The purpose of these lectures is, starting from the classical context, to present the basics of the three fields that lead to an interesting generalisation of the concepts, and to highlight some of the most striking new developments. This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008.     

A parametrized compactness theorem under bounded Ricci curvature

We prove a parametrized compactness theorem on manifolds of bounded Ricci curvature, upper bounded diameter, and lower bounded injectivity radius.     

Ricci curvature and orientability

In this paper we define an orientation of a measured Gromov–Hausdorff limit space of Riemannian manifolds with uniform Ricci bounds from below. This is the first observation of orientability for metric measure spaces. Our orientability has two fundamental properties. One of them is the stability with respect to noncollapsed sequences. As a corollary we see that if the cross section of a tangent cone of a noncollapsed limit space of orientable Riemannian manifolds is smooth, then it is also orientable in the ordinary sense, which can be regarded as a new obstruction for a given manifold to be the cross section of a tangent cone. The other one is that there are only two choices for orientations on a limit space. We also discuss relationships between \(L^2\)-convergence of orientations and convergence of currents in metric spaces. In particular for a noncollapsed sequence, we prove a compatibility between the intrinsic flat convergence by Sormani–Wenger, the pointed flat convergence by Lang–Wenger, and the Gromov–Hausdorff convergence, which is a generalization of a recent work by Matveev–Portegies to the noncompact case. Moreover combining this compatibility with the second property of our orientation gives an explicit formula for the limit integral current by using an orientation on a limit space. Finally dualities between de Rham cohomologies on an oriented limit space are proven.     

A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching

Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.     

Rigidity of shrinking Ricci solitons

We show that a compact Ricci soliton is rigid if and only if the Weyl conformal tensor is harmonic. In the complete noncompact case we prove the same result assuming that the curvature tensor has at most exponential growth and the Ricci tensor is bounded from below.     

Ricci curvature and betti numbers

We derive a uniform bound for the total betti number of a closed manifold in terms of a Ricci curvature lower bound, a conjugate radius lower bound and a diameter upper bound. The result is based on an angle version of Toponogov comparison estimate for small triangles in a complete manifold with a Ricci curvature lower bound. We also give a uniform estimate on the generators of the fundamental group and prove a fibration theorem in this setting.     

Small excess and Ricci curvature

It is proved that a Riemanniann-manifold with Ricci curvature ≥ (n − 1) and a lower injectivity radius bound is a sphere provided the diameter is sufficiently close to π. The author was partially supported by the NSF and the Alfred P. Sloan Foundation.     

Rigidity and Infinitesimal Deformability of Ricci Solitons

In this paper, an obstruction against the integrability of certain infinitesimal solitonic deformations is given. Using this obstruction, we show that the complex projective spaces of even complex dimension are rigid as Ricci solitons although they have infinitesimal solitonic deformations.