Some Type I Solutions of Ricci Flow with Rotational Symmetry

Song, Jian

doi:10.1093/imrn/rnu134

Cited by 6 publications

(15 citation statements)

References 22 publications

(34 reference statements)

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“…where J j , J ∞ are the complex structures on X j , X ∞ , respectively, compatible with the Kähler metrics g j , g ∞ . Since the restriction of the metrics g j to E are (n − 1)g F S where g F S is the Fubini-Study metric on CP n−1 , we have Lemma 2.2 (see also [21]). The diameter of (E, g j | E ) is D n = α n (n − 1) 1/2 , hence uniformly bounded.…”

Section: 3mentioning

confidence: 93%

“…(1) ( [21]) If lim inf t→T (T −t) −1 V ol(X, g(t)) = ∞, then (X, g j (t), p) sub-converges in C ∞ Cheeger-Gromov-Hamilton (CGH) sense to a complete shrinking non-flat gradient Kähler Ricci soliton on a complete Kähler manifold diffeomorphic to C n , for any fixed point p ∈ E, the exceptional divisor.…”

Section: Preliminariesmentioning

confidence: 99%

“…The evolution equations for the potentials of the evolving metrics ω(t) = i∂∂u(ρ, t) for ρ ∈ (−∞, ∞) and t ∈ [0, T ), where T is given in (2.6), are given by ( [25,21])…”

Section: A Priori Estimatesmentioning

confidence: 99%

“…In fact a 0 (n − 1) < b 0 (n + 1) is equivalent to the condition (1.2) . It is then shown in [21] that the flow (1.1) must develop Type I singularities and the parabolic blow up of the Type I Ricci flow along the exceptional divisor converges to a complete non-flat shrinking Kähler Ricci soliton on a complete manifold diffeomorphic to C n blown-up at one point.…”

Section: Introductionmentioning

confidence: 99%

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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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Section: 3mentioning

confidence: 93%

Section: Preliminariesmentioning

confidence: 99%

“…The evolution equations for the potentials of the evolving metrics ω(t) = i∂∂u(ρ, t) for ρ ∈ (−∞, ∞) and t ∈ [0, T ), where T is given in (2.6), are given by ( [25,21])…”

Section: A Priori Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

“…

…”

mentioning

confidence: 99%

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