On the Collapsing Rate of the Kähler–Ricci Flow with Finite-Time Singularity

Fong, Frederick

doi:10.1007/s12220-013-9458-x

Cited by 9 publications

(12 citation statements)

References 18 publications

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“…Our main result is to show that the flow must also develop Type I singularities when it does not collapse and the blow-up limit is a nontrivial complete shrinking gradient Kähler-Ricci soliton. Combined with the results of Zhu [28] and Fong [7], we have the following theorem. Theorem 1.1.…”

Section: Introductionsupporting

confidence: 60%

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Some Type I Solutions of Ricci Flow with Rotational Symmetry

Song

2014

Int Math Res Notices

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We prove that the Ricci flow on CP n blown-up at one point starting with any rotationally symmetric Kähler metric must develop Type I singularities.In particular, if the total volume does not go to zero at the singular time, the parabolic blow-up limit of the Type I Ricci flow along the exceptional divisor is a complete non-flat shrinking gradient Kähler-Ricci soliton on a complete Kähler manifold homeomorphic to C n blown-up at one point.

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Section: Introductionsupporting

confidence: 60%

“…is the standard flat metric on C n−1 and g F S the Fubini-Study metric on CP 1 for any sequence of points p j [7].…”

Section: Introductionmentioning

confidence: 99%

Some Type I Solutions of Ricci Flow with Rotational Symmetry

Song

2014

Int Math Res Notices

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“…(2) ( [9]) If lim inf t→T (T − t) −1 V ol(X, g(t)) ∈ (0, ∞), then (X, g j (t), p j ) sub-converges in C ∞ -CGH sense to (C n−1 × CP 1 , g C n−1 ⊕ (−t)g F S ), where g C n−1 is the standard flat metric on C n−1 and g F S is the Fubini-Study metric on CP 1 for any sequence of points p j . (3) ( [31]) If lim inf t→T (T − t) −1 V ol(X, g(t)) = 0, then (X, g j (t)) converges in C ∞ -CGH sense to the unique compact shrinking Kähler Ricci soliton on CP n blown-up at one point.…”

Section: Preliminariesmentioning

confidence: 99%

“…When the initial Kähler class satisfies a 0 (n − 1) > b 0 (n + 1), the flow collapses to CP n−1 at T = (b 0 − a 0 )/2 ( [25]). It is shown in [9] that the flow must develop Type I singularities and the rescaled flow converges in Cheeger-Gromov-Hamilton sense to the ancient solution that splits isometrically as C n−1 × CP 1 . The initial Kähler class condition of a 0 (n − 1) < b 0 (n + 1), is equivalent to the limiting total volume being strictly positive at the singular time T = a 0 /(n − 1), i.e., (1.2) lim inf t→T − V ol(X, g(t)) > 0, and the flow contracts the exceptional divisor D 0 at T ( [25]).…”

Section: Introductionmentioning

confidence: 99%

On Feldman–Ilmanen–Knopf’s conjecture for the blow-up behavior of the Kähler Ricci flow

Guo

Song

2016

Mathematical Research Letters

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We consider the Ricci flow on CP n blown-up at one point starting with any U (n)invariant Kähler metric. It is proved in [31,9,21] that the Kähler-Ricci flow must develop Type I singularities. We show that if the total volume does not go to zero at the singular time, then any Type I parabolic blow-up limit of the Ricci flow along the exceptional divisor is the unique U (n)-complete shrinking Kähler-Ricci soliton on C n blown-up at one point. This establishes the conjecture of Feldman-Ilmanen-Knopf [8].It is proved in [13,7] that if the Ricci flow develops type I singularity on a closed manifold, then the type I blow-up limit along essential singularities must be a nontrivial complete shrinking Ricci soliton.CP n blown-up at one point is in fact a CP 1 bundle over CP n−1 given byLet D 0 be the exceptional divisor of X defined by the image of the section (1, 0) of O CP n−1 ⊕ O CP n−1 (−1) and D ∞ be the divisor of X defined by the image of the section (0, 1) of O CP n−1 ⊕ It is proved by Feldman-Ilmanen-Knopf [8]) that there exists a unique U (n) invariant complete Kähler-Ricci gradient shrinking soliton on C n blown-up at one point (FIK soliton) and they further made the following conjecture.Conjecture 1.1. Let g(t) be the U (n) invariant metrics satisfying the Kähler-Ricci flow on X = CP n blown-up at one point for t ∈ [0, T ). Let T ∈ (0, ∞) be the singular time and lim inf t→T − V ol(X, g(t)) > 0.Then the flow develops type I singularities and any type I parabolic blow-up limit of g(t) with a fixed base point in the exceptional divisor E is the unique FIK soliton on C n blown-up at one point.This conjecture was partially established by Maximo ([12]) when the dimension n = 2 under certain open conditions on the initial metric. Our main result in this paper is to show that in the non-collapsed case, the blow-up limit of the Kähler Ricci flow is biholomorphic to C n blown-up at one point and the limit Kähler Ricci soliton is the FIK soliton constructed in [8] on C n blown-up at one point, hence establishing Conjecture 1.1. Our main theorem is Theorem 1.2. Let X be CP n blown-up at one point and E be the exceptional divisor. Let g(t) be the U (n)-invariant solution to (1.1) on X on [0, T ). If lim inf

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“…Surprisingly, it is still an open problem to determine the precise behavior of the flow even in the case of P 1 × P 1 when one of the fibers collapses. It was shown by Song-Székelyhidi-Weinkove [59] that the diameter of the collapsing fiber is bounded above by a multiple of (T − t) 1/3 , but this falls short of the optimal rate of (T − t) 1/2 (see also [21]). It is expected that the blow-up limit is a product with a flat direction and this has been proved under symmetry conditions [20,57].…”

Section: Some Open Problemsmentioning

confidence: 99%

The Kähler-Ricci flow on compact Kähler manifolds

Weinkove¹

2016

Geometric Analysis

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These lecture notes are based on five hours of lectures given at the Park City Math Institute in the summer of 2013. The notes are intended to be a leisurely introduction to the Kähler-Ricci flow on compact Kähler manifolds. They are aimed at graduate students who have some background in differential geometry, but do not necessarily have any knowledge of Kähler geometry or the Ricci flow. There are exercises throughout the text. The goal is that by the end, the reader will learn the basic techniques in the Kähler-Ricci flow and know enough to be able to explore the current literature.The material covered by these notes is as follows. In the first lecture, we give a quick introduction to some of the main definitions and tools of Kähler geometry. In Lecture 2, we introduce the Kähler-Ricci flow and give some simple examples, before stating, and in Lecture 3 proving, the maximal existence time theorem for the flow. In Lecture 4 we prove long time convergence results in the cases when the manifold has negative or zero first Chern class. Finally in Lecture 5 we discuss more recent work on the behavior of the flow on Kähler surfaces. We also "go beyond" the Kähler-Ricci flow by discussing a new flow on complex manifolds called the Chern-Ricci flow.The Kähler-Ricci flow started as a small branch of the study of Hamilton's Ricci flow, but by now is itself a vast area of research. As a consequence, we have had to omit many topics. For the interested reader seeking more complete expository sources: the chapter [66] by Jian Song and the author contains many of the results of these notes and much more; the works [4,8,31] (in the same volume as [66]) and the more general survey [53] are excellent sources of information.The author thanks Matt Gill who was the teaching assistant for this course, for his help in writing the exercises. In addition, thanks go to the organizers, Hubert Bray, Greg Galloway, Rafe Mazzeo and Natasa Sesum, of the research program of the 2013 PCMI Summer Session for giving the author the opportunity to participate in this exciting event. Discussions with researchers and graduate students at the Park City Math Institute were invaluable in shaping the form of these notes. The author also thanks Valentino Tosatti for some helpful comments on a previous version of these notes.

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On the Collapsing Rate of the Kähler–Ricci Flow with Finite-Time Singularity

Cited by 9 publications

References 18 publications

Some Type I Solutions of Ricci Flow with Rotational Symmetry

Some Type I Solutions of Ricci Flow with Rotational Symmetry

On Feldman–Ilmanen–Knopf’s conjecture for the blow-up behavior of the Kähler Ricci flow

The Kähler-Ricci flow on compact Kähler manifolds

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