In this paper, we study gradient Ricci expanding solitons (X, g) satisfying Rc = cg +Df, where Rc is the Ricci curvature, c < 0 is a constant, and Df is the Hessian of the potential function f on X . We show that for a gradient expanding soliton (X, g) with non-negative Ricci curvature, the scalar curvature R has at least one maximum point on X , which is the only minimum point of the potential...

The Ricci curvature of graphs, as recently introduced by Lin, Lu, and Yau following a general concept due to Ollivier, provides a new and promising isomorphism invariant. This paper presents a simplified exposition of the concept, including the so-called logistic diagram as a computational or visualization aid. Two new infinite classes of graphs with positive Ricci curvature are identified. A l...

Abstract. Let M be a compact Riemannian manifold and the metrics g = g(t) evolve by the Ricci flow. We prove the following result. The Sobolev imbedding by Aubin or Hebey, perturbed by a scalar curvature term and modulo sharpness of constants, holds uniformly for (M, g(t)) for all time if the Ricci flow exists for all time; and if the Ricci flow develops a singularity in finite time, then the s...

It is proved that every locally conformal flat Riemannian manifold all of whose Jacobi operators have constant eigenvalues along every geodesic is with constant principal Ricci curvatures. A local classification (up to an isometry) of locally conformal flat Riemannian manifold with constant Ricci eigenvalues is given in dimensions 4, 5, 6, 7 and 8. It is shown that any n-dimensional (4 ≤ n ≤ 8)...

In earlier work, carrying out numerical simulations of the Ricci flows of families of rotationally symmetric geometries on S3, we have found strong support for the contention that (at least in the rotationally symmetric case) the Ricci flow for a “critical” initial geometry– one which is at the transition point between initial geometries (on S3) whose volume-normalized Ricci flows develop a sin...

Let (M, g) be a compact n-dimensional (n 2) manifold with nonnegative Ricci curvature, and if n 3, then we assume that (M, g) × R has nonnegative isotropic curvature. The lower bound of the Ricci flow’s existence time on (M, g) is proved. This provides an alternative proof for the uniform lower bound of a family of closed Ricci flows’ maximal existence times, which was first proved by E. Cabeza...

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80's, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on co...

The Ricci flow is an evolution system on metrics. For a given metric as initial data, its local existence and uniqueness on compact manifolds was first established by Hamilton [8]. Later on, De Turck [4] gave a simplified proof. In the later of 80's, Shi [20] generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on co...