A uniqueness result of Kähler Ricci flow with an application

Fan, Xu-Qian

doi:10.1090/s0002-9939-06-08510-8

Cited by 5 publications

(6 citation statements)

References 11 publications

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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(t i ) for all i. So γ(q i ) is also a zero of ∇ i R (i) .…”

Section: Long Time Solutions and Local Limitsmentioning

confidence: 91%

“…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 Kähler-Ricci flow (see [18,21]), F is an holomorphic isometry with respect to g(t) for all t. As a mapping on M , F (x, ỹ) = (f 1 (x, ỹ), f 2 (x, ỹ)) where…”

Section: Fiber Bundle Structuresmentioning

confidence: 99%

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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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“…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(t i ) for all i. So γ(q i ) is also a zero of ∇ i R (i) .…”

Section: Long Time Solutions and Local Limitsmentioning

confidence: 91%

“…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%

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On the simply connectedness of nonnegatively curved Kähler manifolds and applications

Chau

Tam

2011

Trans. Amer. Math. Soc.

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“…For the Kähler-Ricci flow, it is more easy to obtain uniqueness. The following is a generalization of the result in [9]. Here we do not assumption the curvature is bounded, and we do not assume that the Ricci form has a potential, which is assumed in [9].…”

Section: Uniquenessmentioning

confidence: 95%

Longtime existence of the Kähler-Ricci flow on ℂⁿ

Chau

Tam

2017

Trans. Amer. Math. Soc.

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We produce longtime solutions to the Kähler-Ricci flow for complete Kähler metrics on C n without assuming the initial metric has bounded curvature, thus extending results in [3]. We prove the existence of a longtime bounded curvature solution emerging from any complete U (n)-invariant Kähler metric with non-negative holomorphic bisectional curvature, and that the solution converges as t → ∞ to the standard Euclidean metric after rescaling. We also prove longtime existence results for more general Kähler metrics on C n which are not necessarily U (n)-invariant.

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“…In higher dimensions, Fan [14] studied the uniqueness and convergence of U (n)-invariant Kähler-Ricci flow on C n with positive bisectional curvature. However, his result assumes upper bounds on curvatures and relies on the short time existence theorem of Shi [34] and some earlier convergence results of Chau and Tam [6].…”

Section: The U (N)-invariant Kähler-ricci Flow Equationmentioning

confidence: 99%

$U(n)$-invariant Kähler–Ricci flow with non-negative curvature

Yang

Zheng

2013

Communications in Analysis and Geometry

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It is interesting to know the existence of the Kähler-Ricci flow on complete non-compact Kähler manifolds with non-negative holomorphic bisectional curvature. In this paper, we study U (n)invariant Kähler-Ricci flow on C n with non-negative curvature. Motivated by the recent work of Wu and the second named author [37], we also study examples of U (n)-invariant complete Kähler metrics on C n with positive and unbounded curvature.252 Bo Yang & Fangyang Zheng 4 Discussions on the existence of U (n)-invariant Kähler-Ricci flow 283 4.1 Theorem of Cabezas-Rivas and Wilking 283 4.2 Ricci flow on double covers with rotational symmetry 283 4.3 Conditions that ensure the existence of Kähler-Ricci flow 285 Acknowledgements 290 References 290

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A uniqueness result of Kähler Ricci flow with an application

Abstract: Abstract. In this paper, we will study the problem of uniqueness of Kähler Ricci flow on some complete noncompact Kähler manifolds and the convergence of the flow on C n with the initial metric constructed by Wu and Zheng.

Cited by 5 publications

References 11 publications

On the simply connectedness of nonnegatively curved Kähler manifolds and applications

On the simply connectedness of nonnegatively curved Kähler manifolds and applications

Longtime existence of the Kähler-Ricci flow on ℂⁿ

$U(n)$-invariant Kähler–Ricci flow with non-negative curvature

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