In this paper, we show that any ancient solution to the Ricci flow with the reduced volume whose asymptotic limit is sufficiently close to that of the Gaussian soliton is isometric to the Euclidean space for all time. This is a generalization of Anderson's result for Ricciflat manifolds. As a corollary, a gap theorem for gradient shrinking Ricci solitons is also obtained.We say that (M, g(τ )) is ancient when g(τ ) exists for all τ ∈ [0, ∞). Ancient solutions are important objects in the study of singularities of the Ricci flow. The limit V(g) := lim τ →∞Ṽ(p,0) (τ ) will be called the asymptotic reduced volume of the flow g(τ ). We will see in Lemma 3.1 below that V(g) is independent of the choice of p ∈ M .By regarding a Ricci-flat metric as an ancient solution as in Theorem 1.1, we recover the following result, which is the motivation of the present paper.There exists ε n > 0 which satisfies the following: let (M n , g) be an n-dimensional complete Ricci-flat Riemannian manifold. Suppose that the asymptotic volume ratio ν(g) := lim r→∞ Vol B(p, r)/ω n r n of g is greater than 1 − ε n . Here ω n stands for the volume of the unit ball in the Euclidean space (On the way to the proof of Theorem 1.1, we establish several lemmas. Here we state one of them as a theorem, which is of independent interest. Theorem 1.3. Let (M n , g(τ )), τ ∈ [0, ∞) be a complete ancient solution to the Ricci flow on M with Ricci curvature bounded below. If V(g) > 0, then the fundamental group of M is finite. In particular, any ancient κ-solution to the Ricci flow has finite fundamental group.
Abstract. Distance functions of metric spaces with lower curvature bound, by definition, enjoy various metric inequalities; triangle comparison, quadruple comparison and the inequality of Lang-Schroeder-Sturm. The purpose of this paper is to study the extremal cases of these inequalities and to prove rigidity results. The spaces which we shall deal with here are Alexandrov spaces which possibly have infinite dimension and are not supposed to be locally compact.
In this paper, we consider the behavior of the total absolute and the total curvature under the Ricci flow on complete surfaces with bounded curvature. It is shown that they are monotone non-increasing and constant in time, respectively, if they exist and are finite at the initial time. As a related result, we prove that the asymptotic volume ratio is constant under the Ricci flow with non-negative Ricci curvature, at the end of the paper.
Abstract. In this paper, we consider two different monotone quantities defined for the Ricci flow and show that their asymptotic limits coincide for any ancient solutions. One of the quantities we consider here is Perelman's reduced volume, while the other is the local quantity discovered by Ecker, Knopf, Ni and Topping. This establishes a relation between these two monotone quantities.
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