Properties Of The Limit Of The Ricci Flow

In 1982, Ricci flow was introduced in Hamilton's seminal paper [47]. Since then, Ricci flow has become a powerful tool to study Riemannian geometry. Fol-lowing Hamilton's program, G. Perelman recently solved the Poincare conjecture, which proposes a topological characterization of the 3-sphere, and Thurston's ge-ometrization conjecture [79,80,81]. Besides this, Ricci flow has been used to solve several other important problems, such as a new proof of the uniformization of Riemannian surfaces [16,49,23], the classification of manifolds with positive curvature operator [2], and the 1/4-pinching theorem [1], etc.In Perelman's paper [79], there are many claims without detailed proofs. For example, Corollary 9.3, Ni first gave its detailed proof in his paper [77] using the properties of Perelman's reduced distance. Recently, Chau, Tarn and Yu [29] also gave another detailed proof using estimates of fundamental solutions. On the other hand, Ni has proved a similar result for closed manifolds with nonnegative Ricci curvature in 2004. I would like to unify the two inequalities. In the chapter 3 of this thesis, following Chau, Tam and Yu's argument, we obtain our first main theorem.Theorem A. Let (M, g(t)) be a solution of the super Ricci flow, i.e.,(?)= hij≤Rij,τ= T-t, on M x [0. T] for some T>0, satisfying assumptions where H=gijhij, and bounded |▽kRm| and |▽kh| for k=0,1,2. Let p∈M be a fixed point, and let Z(p,T;x,t) be a positive fundamental solution of the conjugate heat equation centered at (p,T), i.e.,limt→TZ(P,T;x,t)=δp. Let u(x,t)= Z(p,T;x,t) andThe key points of this proof are the same estimates of fundamental solutions under super Ricci flow with above assumptions and a nice "monotony formula"In the chapter 4 of this thesis, we state the second main theorem, which gives a result about the convergence of fundamental solutions in the Cheeger-Gromov sense under Ricci flow, which was claimed by Perelman [79]. It plays an important role in the proof of Perelman's pseudolocality theorem.Theorem B. Let (Mk,gk(t),xk) be a sequence of pointed Ricci flow (?)gij-2Rij, where each Mk is a non-compact manifold with bounded curvature such that (Mk,gk(t)) is complete for t∈[0,T]. Suppose for some constant K>0 and converges in the C∞pointed Cheeger-Gromov sense to a smooth solution of Ricci flow,Define the function uk on Mk×[0,T) to be the minimal positive fundamental solution of the conjugate heat equation, (-(?)-△+Rk)u=0, limiting to theδ-function centered at xk as time approaches T; i.e., uk is the minimal positive solution to (-(?)-△+Rk)u=0 andThenΦk*(uk) subconverges uniformly on every compact subset of M∞x (0,T) to the minimal positive fundamental solution u of the conjugate heat equation on M∞×(0,T) limiting to theδ-function centered at p as time approaches T.The key point is to use some estimates of fundamental solutions proved in [29]. In the chapter 5 of this thesis,we obtain several theorems.Comparing Chen's global lower bound of the scalar curvature for Ricci flow,we obtain a local lower bound of the scalar curvature for Ricci flow,using Yokota's argument,[99].It is our third main theorem.Theorem C.For any 0<ε<2/n.Suppose(Mn,g(t)), t∈[α,β] is a complete solution to Ricci flow,p∈M,then there exist constants C(p)depending on p and the metrics g(t),t∈[α,β] and C such that when c≥C(p),we have whenever x∈Bg(t)(p,c),t∈(a,β],where A(ε)=2/n-ε, B(ε)=3C/2√Aεc2 .It implies two corollaries.Corollary C1.Suppose(Mn,g(t)), t∈[a,β],is a complete solution to the Ricci flow,then on M×(α,β]Since ancient solution is a special solution to the Ricci flow for t∈(-∞,0]. Then we have the following property.Corollary C2.If(Mn,g(t)), t∈(-∞,0],is a complete ancient solution to the Ricci flow,then on M×(-∞,O].From now on,we study some properties of gradient Ricci solitions. The properties of Ricci solitons have been studied by many people.We say that a quadruple(Mn,g,f,ε),where(Mn,g)is a Riemannian main-fold,f is a smooth function on Mn andε∈R,is a gradient Ricci soliton if We call f the potential function.We say that g is shrinking,steady,or expanding ifε<0,ε=0,orε>0,respectively. The following property is a consequence of Chen's results in [34], but I will give the direct proof using Ricci solitons equation.Theorem D. Suppose (Mn. g, f,ε) be a noncompact complete gradient Ricci soliton. Then(1) If the gradient soliton is shrinking, then R≥0. Moreover, if the scalar curvature obtains 0 at some point, then (Mn,g) is isometric to Rn.(2) If the gradient soliton is steady, then R≥0. Moreover, if the scalar curvature obtains 0 at some point, then (Mn,g) is Ricci flat.(3) If the gradient soliton is expanding, then R≥-nε/2. Moreover, if the scalar curvature obtains -nε/2 at some point, then (Mn,g) is Einstein.The shrinkers have at most Euclidean volume growth was obtained by Cao and Zhou [32] and Munteanu [66]. Add some assumptions about the scalar cur-vature, we also obtain the shrinkers have at most rσ(σ< n) volume growth as our fourth main theorem.Theorem E. Let (Mn,g,f,-1) be a complete shrinking gradient Ricci soliton with R≥δ>0. Then given o∈Mn, there exists a constant C<∞depending only onδ, o and the soliton such that for all r≥0.Then we obtain the result which was proved by Carrillo and Ni [26].Corollary E1. Any nonflat shrinking gradient Ricci soliton with nonnega-tive Ricci curvature must have V(g)=0.Note that V(g) is defined on manifold with nonnegative Ricci curvature asThe classification of three dimensional shrinking gradient Ricci solitons has been solved after many people's work. Our the last main theorem is the only condition in a result was proved by Petersen and Wylie [83] with vanishing Weyl tensor. Theorem F. Let (M,g,f,-1) be a complete shrinking gradient Ricci soli-ton, then Moreover,Corollary F1. Let (M,g,f,-1) be a complete shrinking gradient Ricci soliton of dimension n≥3 with W=0, then M is a finite quotient of Sn,Rn or Sn-1×R. In particular, a three dimensional shrinking gradient Ricci soliton is a finite quotient of S3, R3 or S2×R.