Lower Ricci Curvature Research Articles

Lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature

We present some new lower bound estimates of the first eigenvalue for compact manifolds with positive Ricci curvature in terms of the diameter and the lower Ricci curvature bound of the manifolds. For compact manifolds with boundary, it is assumed that, with respect to the outward normal, it is of nonnegative second fundamental form for the first Neumann eigenvalue and the mean curvature of the boundary is nonnegative for the first Dirichlet eigenvalue.

Ricci Curvature and Volume Convergence

The purpose of this paper is to give a new (integral) estimate of distances and angles on manifolds with a given lower Ricci curvature bound. This will provide us with an integral version of the Toponogov comparison triangle theorem for Ricci curvature and extreme triangles (see the earlier works [Cl] and [C2] for an analog of this when the manifold has positive Ricci curvature). Using this, we prove the Anderson-Cheeger conjecture saying that the volume is a continuous function on the space of all closed n-manifolds with Ricci curvature greater or equal to -(n - 1) equipped with the GromovHausdorff metric. We also prove Gromov's conjecture (for n 57 3) saying that an almost nonnegatively Ricci curved n-manifold with first Betti number equal to n is a torus. Further, we prove a conjecture of Anderson-Cheeger saying that an open n-manifold with nonnegative Ricci curvature whose tangent cone at infinity is in is itself in. Finally we prove a conjecture of Fukaya-Yamaguchi. We will now describe these results in more detail. Let dGH denote the Gromov-Hausdorff distance [GLP]. First we have the following result which was conjectured by Anderson-Cheeger.

Orbifolds with lower Ricci curvature bounds

We show that the first betti number $b_1(O)=\dim H_1(O,{\mathbb R})$ of a compact Riemannian orbifold $O$ with Ricci curvature $\mathrm {Ric}(O)\ge -(n-1)k$ and diameter $\operatorname {diam}(O)\le D$ is bounded above by a constant $c(n,kD^2)\ge 0$, depending only on dimension, curvature and diameter. In the case when the orbifold has nonnegative Ricci curvature, we show that the $b_1(O)$ is bounded above by the dimension $\dim O$, and that if, in addition, $b_1(O)=\dim O$, then $O$ is a flat torus $T^n$.

On Function Theory on Spaces with a Lower Ricci Curvature Bound

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

Maximum principle for hypersurfaces

We give an intrinsic proof and a generalization of the interior and boundary maximum principle for hypersurfaces in Riemannian and Lorentzian manifolds. Moreover, we show some new applications to manifolds with lower Ricci curvature bounds. E.g. we prove a local and a Lorentzian version of Cheng's maximal diameter theorem and a non-existence result for closed minimal hypersurfaces.

Displacement convexity of Boltzmann’s entropy characterizes the strong energy condition from general relativity

On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation. These notions also make sense in a (nonsmooth) metric measure setting, where they have found powerful applications. This article initiates the development of an analogous theory for lower Ricci curvature bounds in timelike directions on a (globally hyperbolic) Lorentzian manifold. In particular, we lift fractional powers of the Lorentz distance (a.k.a. time separation function) to probability measures on spacetime, and show the strong energy condition of Hawking and Penrose is equivalent to geodesic convexity of the Boltzmann-Shannon entropy there. This represents a significant first step towards a formulation of the strong energy condition and exploration of its consequences in nonsmooth spacetimes, and hints at new connections linking the theory of gravity to the second law of thermodynamics.

Lower diameter bounds for compact shrinking ricci solitons

It is shown that the diameter of a compact shrinking Ricci soliton has a universal lower bound. This is proved by extending universal estimates for the first non-zero eigenvalue of Laplacian on compact Riemannian manifolds with lower Ricci curvature bound to a twisted Laplacian on compact shrinking Ricci solitons.

Ricci curvature and eigenvalue estimate on locally finite graphs

We give a generalizations of lower Ricci curvature bound in the framework of graphs. We prove that the Ricci curvature in the sense of Bakry and Emery is bounded below by $-1$ on locally finite graphs. The Ricci flat graph in the sense of Chung and Yau is proved to be a graph with Ricci curvature bounded below by zero. We also get an estimate for the eigenvalue of Laplace operator on finite graphs: $$\lambda\ge {1\over d D(\exp( d D+1)-1)},$$ where $d$ is the weighted degree of $G$, and $D$ is the diameter of $G$.