Perelman's reduced volume and a gap theorem for the Ricci flow
Communications in analysis and geometry
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
... bjects 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. Theorem 1.2 [1, Gap Lemma 3.1]. 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 (R n , g E ). Then (M n , g) is isometric to (R n , g E ). 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.