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Register a new userThe Ricci Flow on Complete Noncompact K{a}hler Manifolds
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Xi-Ping Zhu
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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.
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We prove that a complete noncompact K{a}hler manifold $M^{n}$of positive bisectional curvature satisfying suitable growth conditions is biholomorphic to a pseudoconvex domain of {bf C}$^{n}$ and we show that the manifold is topologically {bf R}$^{2n}$. In particular, when $M^{n}$ is a K{a}hler surface of positive bisectional curvature satisfying certain natural geometric growth conditions, it is biholomorphic to {bf C}$^{2}$.
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.
In this paper, we study the existence of Poisson metrics on flat vector bundles over noncompact Riemannian manifolds and discuss related consequence, specially on the applications in Higgs bundles, towards generalizing Corlette-Donaldson-Hitchin-Simpsons nonabelian Hodge correspondence to noncompact K{a}hler manifolds setting.
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 $ktoinfty$.
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.
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