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Register a new userUniqueness of the Ricci Flow on Complete Noncompact Manifolds
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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 cite{Ha1}. Later on, De Turck cite{De} gave a simplified proof. In the later of 80s, Shi cite{Sh1} generalized the local existence result to complete noncompact manifolds. However, the uniqueness of the solutions to the Ricci flow on complete noncompact manifolds is still an open question. Recently it was found that the uniqueness of the Ricci flow on complete noncompact manifolds is important in the theory of the Ricci flow with surgery. In this paper, we give an affirmative answer for the uniqueness question. More precisely, we prove that the solution of the Ricci flow with bounded curvature on a complete noncompact manifold is unique.
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In this paper we survey the recent developments of the Ricci flows on complete noncompact K{a}hler manifolds and their applications in geometry.
In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let $g(t)$ be a smooth complete solution to the Ricci flow on $mathbb{R}^{3}$, with the canonical Euclidean metric $E$ as initial data, then $g(t)$ is trivial, i.e. $g(t)equiv E$.
We consider the Ricci flow $frac{partial}{partial t}g=-2Ric$ on the 3-dimensional complete noncompact manifold $(M,g(0))$ with non-negative curvature operator, i.e., $Rmgeq 0, |Rm(p)|to 0, ~as ~d(o,p)to 0.$ We prove that the Ricci flow on such a manifold is nonsingular in any finite time.
We verify a conjecture of Perelman, which states that there exists a canonical Ricci flow through singularities starting from an arbitrary compact Riemannian 3-manifold. Our main result is a uniqueness theorem for such flows, which, together with an earlier existence theorem of Lott and the second named author, implies Perelmans conjecture. We also show that this flow through singularities depends continuously on its initial condition and that it may be obtained as a limit of Ricci flows with surgery. Our results have applications to the study of diffeomorphism groups of three manifolds --- in particular to the Generalized Smale Conjecture --- which will appear in a subsequent paper.
The main result of this paper shows that, if $g(t)$ is a complete non-singular solution of the normalized Ricci flow on a noncompact 4-manifold $M$ of finite volume, then the Euler characteristic number $chi(M)geq0$. Moreover, $chi(M) eq 0$, there exist a sequence times $t_ktoinfty$, a double sequence of points ${p_{k,l}}_{l=1}^{N}$ and domains ${U_{k,l}}_{l=1}^{N}$ with $p_{k,l}in U_{k,l}$ satisfying the followings: [(i)] $dist_{g(t_k)}(p_{k,l_1},p_{k,l_2})toinfty$ as $ktoinfty$, for any fixed $l_1 eq l_2$; [(ii)] for each $l$, $(U_{k,l},g(t_k),p_{k,l})$ converges in the $C_{loc}^infty$ sense to a complete negative Einstein manifold $(M_{infty,l},g_{infty,l},p_{infty,l})$ when $ktoinfty$; [(iii)] $Vol_{g(t_{k})}(Mbackslashbigcup_{l=1}^{N}U_{k,l})to0$ as $ktoinfty$.
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