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

Guo, Bin; Song, Jian

doi:10.4310/mrl.2016.v23.n6.a6

Cited by 4 publications

(10 citation statements)

References 24 publications

(67 reference statements)

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“…As a corollary to the previous works of Zhu [43], Weinkove-Song [36], Fong [18], Guo-Song [21] on Kähler-Ricci flow on Hirzebruch manifolds with Calabi's symmetry, we exhibit various pinching behaviors along the Kähler-Ricci flow when the initial metrics are chosen from examples constructed in Theorem 1.3, in particular we have the following: The above corollary entails the following question: Can we construct a suitable one-parameter family of deformation of Kähler metrics M n,k so that the holomorphic pinching constant is monotone along the deformation?…”

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confidence: 68%

“…By plugging c ≤ n k − 2ǫ p and µ = c p + δ 2 into the above one can conclude that is a sufficient condition for C > 0 is (21) δ 2 < n(2p + 2k + 1) kp(2p + 2k − 1) .…”

Section: Note Thatmentioning

confidence: 99%

“…In other words, for any c ∈ (0, n k − 2ǫ p ], it is easy to pick δ 1 and δ 2 such that all the inequalities in (17), (21), and (23) where A, B, C are given in (18), (19), and (20).…”

Section: Note Thatmentioning

confidence: 99%

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Hirzebruch manifolds and positive holomorphic sectional curvature

Yang¹,

Zheng²

2019

Annales De l'Institut Fourier

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This paper is the first step in a systematic project to study examples of Kähler manifolds with positive holomorphic sectional curvature (H > 0). Previously Hitchin proved that any compact Kähler surface with H > 0 must be rational and he constructed such examples on Hirzebruch surfaces M 2,k = P(H k ⊕ 1 CP 1 ). We generalize Hitchin's construction and prove that any Hirzebruch manifold M n,k = P(H k ⊕ 1 CP n−1 ) admits a Kähler metric of H > 0 in each of its Kähler classes. We demonstrate that the pinching behaviors of holomorphic sectional curvatures of new examples differ from those of Hitchin's which were studied in the recent work of Alvarez-Chaturvedi-Heier. Some connections to recent works on the Kähler-Ricci flow on Hirzebruch manifolds are also discussed.It seems interesting to study the space of all Kähler metrics of H > 0 on a given Kähler manifold. We give higher dimensional examples such that some Kähler classes admit Kähler metrics with H > 0 and some do not.It is the Riemannian sectional curvature restricted on any J-invariant real 2-plane (p165 [29]). Compact Kähler manifolds with positive holomorphic sectional (H > 0) form an interesting class of Kähler manifolds. For example, these manifolds are simply-connected, by the work of Tsukamoto [38]. An averaging argument due to Berger [6] showed that H > 0 implies positive scalar curvature, which further leads to the vanishing of its pluri-canonical ring by a Bochner-Kodaira type identity, see [26]. In 1975 Hitchin [24] proved that any Hirzebruch surface M 2,k = P(H k ⊕ 1 CP 1 ) admits a Kähler metrics with H > 0. Moreover, he proved that any rational surface admits a Kähler metric with positive scalar curvature.

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supporting

confidence: 68%

“…By plugging c ≤ n k − 2ǫ p and µ = c p + δ 2 into the above one can conclude that is a sufficient condition for C > 0 is (21) δ 2 < n(2p + 2k + 1) kp(2p + 2k − 1) .…”

Section: Note Thatmentioning

confidence: 99%

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Hirzebruch manifolds and positive holomorphic sectional curvature

Yang¹,

Zheng²

2019

Annales De l'Institut Fourier

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“…This characterises the Feldman-Ilmanen-Knopf shrinking soliton as the unique shrinking soliton modelling finite time Type I non-collapsed singularities of the Kähler-Ricci flow on compact Kähler surfaces. The "if" direction of Theorem B is known to hold true for U (n)-invariant Kähler-Ricci flows on the blowup of P n at one point [GS16]. Moreover, on this manifold, it is known that any U (n)-invariant solution of the Kähler-Ricci flow developing a finite time singularity is a singularity of Type I [Son15].…”

Section: Resultsmentioning

confidence: 99%

On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

Cifarelli¹,

Conlon²,

Deruelle³

2022

Preprint

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We show that the underlying complex manifold of a complete non-compact twodimensional shrinking gradient Kähler-Ricci soliton (M, g, X) with soliton metric g with bounded scalar curvature Rg whose soliton vector field X has an integral curve along which Rg → 0 is biholomorphic to either C × P 1 or to the blowup of this manifold at one point. Assuming the existence of such a soliton on this latter manifold, we show that it is toric and unique. We also identify the corresponding soliton vector field. Given these possibilities, we then prove a strong form of the Feldman-Ilmanen-Knopf conjecture for finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces, leading to a classification of the bubbles of such singularities in this dimension.(i) M is biholomorphic to either C × P 1 or to Bl p (C × P 1 ), that is, the blowup of C × P 1 at a fixed point p of the standard torus action on C × P 1 . (ii) There exists a biholomorphism γ : M → M such that γ −1 * (JX) lies in the Lie algebra of the real torus T acting on these spaces in the standard way and γ * g is T-invariant. (iii) γ −1 * (JX) is determined and its flow generates a holomorphic isometric S 1 -action of (M, J, γ * g). (iv) Assuming existence, γ * g is the unique T-invariant complete shrinking gradient Kähler-Ricci soliton on M .Conclusions (ii)-(iv) for M = C × P 1 have already been established in [Cif20] where it is shown that any complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on this manifold is isometric to the Cartesian product of the flat Gaussian soliton ω C on C and twice the Fubini-Study metric ω P 1 on P 1 . The new possibility arising is when M is the blowup of C × P 1 at one point, in which case γ −1 * (JX) is given by (2.16). In light of this, we make the following conjecture. Conjecture 1.1. There exists a complete shrinking gradient Kähler-Ricci soliton ω on Bl p (C × P 1 ), that is, the blowup of C × P 1 at a fixed point p of the standard torus action on C × P 1 , invariant under the real torus action induced by the standard real torus action on C × P 1 , with bounded scalar

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“…In four dimensions and higher, this pinching fails in full generality. Indeed, numerous authors have constructed higher-dimensional Ricci flows with finite-time singularities whose blow-up limits do not have non-negative Riemann curvature (see, for example, [7], [26] and [39]). However, if we make the powerful assumption that our Ricci flow is rotationally-invariant, then we can recover the Hamilton-Ivey pinching property.…”

Section: Hamilton-ivey Pinchingmentioning

confidence: 99%

Canonical surgeries in rotationally invariant Ricci flow

Buttsworth¹,

Hallgren²

2022

Preprint

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We construct a rotationally invariant Ricci flow through surgery starting at any closed rotationally invariant Riemannian manifold. We demonstrate that a sequence of such Ricci flows with surgery converges to a Ricci flow spacetime in the sense of [32]. Results of and Haslhofer [29] then guarantee the uniqueness and stability of these spacetimes given initial data. We simplify aspects of this proof in our setting, and show that for rotationally invariant Ricci flows, the closeness of spacetimes can be measured by equivariant comparison maps. Finally we show that the blowup rate of the curvature near a singular time for these Ricci flows is bounded by the inverse of remaining time squared. 8 Construction of the comparison domain 9 Construction of the comparison map 10 Uniqueness of singular Ricci Flows 11 Number of bumps in a Ricci flow spacetime 12 Bounding the blowup rate 13 Appendix: Preparatory results in Section 8 of the Bamler-Kleiner paperProof. It is clear that the manifolds described in Definition 2.1 are all orientable, and that the actions of O(n + 1) on these manifolds are all faithful and proper, with principal isotropy subgroup given by O(n).On the other hand, suppose that the connected Riemannian manifold (M n+1 , g) admits a proper and faithful cohomogeneity one action of the group G = O(n + 1), which acts via isometries, and has H = O(n) as the principal isometry subgroup. Let Mreg denote the set of all points whose isotropy subgroups are (conjugacy equivalent to) H. This is an open subset of M , and M * reg = Mreg/O(n + 1) is a smooth one-dimensional manifold, and comes equipped with a Riemannian metric induced by g. The space M *reg is open and dense in M * = M/O(n + 1), which is therefore a smooth one-dimensional manifold with boundary. We split into cases according to the topological type of M * .Case One: M * = R. In this case, M * reg = M * and since R is contractible, M must appear as the trivial bundle R × G/H = R × S n .Case Two: M * = S 1 . In this case, we again have M * reg = M * . Choose any point p * ∈ M/O(n + 1), and consider M \ π −1 (p * ). Then like in Case One, this manifold appears as R × S n . The original manifold is then found by identifying the copies of S n at their two ends. The are only two O(n + 1)-invariant diffeomorphisms on S n , namely, ±I, so there are only two possibilities: S 1 × S n , or a non-trivial S 1 bundle, but this second example is not oriented.Case Three: M * = [0, ∞). Let K be the isotropy of the single non-principal orbit. It is well known that K/H is a sphere, say, S d , and K acts linearly on R d+1 , and transitively on S d . Since H = O(n), it is clear that K = O(n + 1) = G, so that the singular orbit is just a point. The resulting manifold is R n+1 .Case Four: M * = [0, 1]. This is just a gluing of two of the manifolds from Case Three along a principal orbit; the result is the compact S n+1 .It is convenient to note that g|Mreg can always be expressed as a warped Riemannian product.

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

Cited by 4 publications

References 24 publications

Hirzebruch manifolds and positive holomorphic sectional curvature

Hirzebruch manifolds and positive holomorphic sectional curvature

On finite time Type I singularities of the Kähler-Ricci flow on compact Kähler surfaces

Canonical surgeries in rotationally invariant Ricci flow

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