Uniqueness of the Ricci flow on complete noncompact manifolds

Chen, Bing-Long; Zhu, Xi-Ping

doi:10.4310/jdg/1175266184

Cited by 148 publications

(182 citation statements)

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“…In this way, g(t) is homogeneous and so complete and of bounded curvature for all t; hence the uniqueness also follows from [17]. It is actually a simple matter to prove that the uniqueness, in turn, implies our assumption of (R n , · μ0 )-invariance.…”

Section: Ricci Flow Starting At a Metric G µmentioning

confidence: 93%

The Ricci flow for simply connected nilmanifolds

Lauret

2011

Communications in Analysis and Geometry

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We prove that the Ricci flow g(t) starting at any metric on R n that is invariant by a transitive nilpotent Lie group N can be obtained by solving an ordinary differential equation (ODE) for a curve of nilpotent Lie brackets on R n . By using that this ODE is the negative gradient flow of a homogeneous polynomial, we obtain that g(t) is type-III, and, up to pull-back by time-dependent diffeomorphisms, that g(t) converges to the flat metric, and the rescaling |R(g(t))| g(t) converges to a Ricci soliton in C ∞ , uniformly on compact sets in R n . The Ricci soliton limit is also invariant by some transitive nilpotent Lie group, though possibly nonisomorphic to N .

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Section: Ricci Flow Starting At a Metric G µmentioning

confidence: 93%

The Ricci flow for simply connected nilmanifolds

Lauret

2011

Communications in Analysis and Geometry

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“…By the uniqueness of Kähler-Ricci flow [18] (see also [21]), γ is also an isometry of g(t) and hence of a i g(…”

Section: Theorem 24 Let (M G(t)) Be As In the Basic Assumption 1 mentioning

confidence: 99%

“…Let F ∈ Γ, the first fundamental group of M with respect to the metric g(0). By the uniqueness of the Kähler-Ricci flow (see [18,21]), F is a holomorphic isometry with respect to g(t) for all t. As a mapping onM ,…”

Section: Fiber Bundle Structuresmentioning

confidence: 99%

On the simply connectedness of nonnegatively curved Kähler manifolds and applications

Chau

Tam

2011

Trans. Amer. Math. Soc.

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Abstract. We study complete noncompact long-time solutions (M, g(t)) to the Kähler-Ricci flow with uniformly bounded nonnegative holomorphic bisectional curvature. We will show that when the Ricci curvature is positive and uniformly pinched, i.e. R ij ≥ cRg ij at (p, t) for all t for some c > 0, then there always exists a local gradient Kähler-Ricci soliton limit around p after possibly rescaling g(t) along some sequence t i → ∞. We will show as an immediate corollary that the injectivity radius of g(t) along t i is uniformly bounded from below along t i , and thus M must in fact be simply connected. Additional results concerning the uniformization of M and fixed points of the holomorphic isometry group will also be established. We will then consider removing the condition of positive Ricci curvature for (M, g(t)). Combining our results with Cao's splitting for Kähler-Ricci flow (2004) and techniques of Ni and Tam (2003), we show that when the positive eigenvalues of the Ricci curvature are uniformly pinched at some point p ∈ M , then M has a special holomorphic fiber bundle structure. We will treat as special cases, complete Kähler manifolds with nonnegative holomorphic bisectional curvature and average quadratic curvature decay as well as the case of steady gradient Kähler-Ricci solitons.

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“…He would like to thank Prof. Luen-Fai Tam for his inspired guidance, constant encouragement and very worthy advice. He would also like to thank the referee for pointing out a gap in Proposition 3.1, for giving many constructive suggestions that will make the presentation clearer, and for supplying the information on the paper [5].…”

Section: Acknowledgmentsmentioning

confidence: 99%

“…Recently, Chen and Zhu [5] proved independently the uniqueness result for the Ricci flow up to some time T assuming only the curvature is bounded at each time…”

Section: Introductionmentioning

confidence: 99%