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

Abstract

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

On Kähler Ricci shrinker surfaces

“…Then there are two possibilities: either along each flow line, scalar curvature tends to zero, or there is a flow line along which scalar curvature tends to a positive number. Combining the curvature estimate (1.4) with Kähler condition, Cifarelli-Conlon-Deruelle [24] shows that the aforementioned dichotomy can be expressed as a simpler one: either lim This is proved by Conlon-Deruelle-Sun (cf. [25]).…”

“…This is finished in Cifarelli-Conlon-Deruelle [24] and Bamler-Cifarelli-Conlon-Deruelle [5]. In Cifarelli-Conlon-Deruelle [24], it is shown that M is biholomorphic to P 1 ×C or Bl 1 (P 1 ×C).…”

“…This is finished in Cifarelli-Conlon-Deruelle [24] and Bamler-Cifarelli-Conlon-Deruelle [5]. In Cifarelli-Conlon-Deruelle [24], it is shown that M is biholomorphic to P 1 ×C or Bl 1 (P 1 ×C). In the former case, they also show that the metric must be isometric to the standard P 1 × C. In the latter case, they prove uniqueness and conjecture the existence (cf.…”

“…In the former case, they also show that the metric must be isometric to the standard P 1 × C. In the latter case, they prove uniqueness and conjecture the existence (cf. Conjecture 1.1 of [24]). In [5], Bamler-Cifarelli-Conlon-Deruelle study a blowup limit of a Kähler Ricci flow with toric symmetry, which must be a noncompact Kähler Ricci shrinker.…”

An Aubin continuity path for shrinking gradient Kähler-Ricci solitons

“…The principal motivation behind Theorem A is that it provides a first step in the construction, via the continuity method, of a complete shrinking gradient Kähler-Ricci soliton with bounded scalar curvature on the blowup of C × P 1 at one point, a manifold identified in [CCD22, Conjecture 1.1] as a potential new example admitting such a soliton. As shown in [CCD22], a soliton on this manifold with bounded scalar curvature must be invariant under the standard torus action, and even though the conditions on the potential soliton vector field X of Theorem A are stringent, they are evidently true for the manifold in question as one can check in [CCD22,Example 2.32]. This example therefore fits into the framework of Theorem A, precisely with D = P 1 and π the blowup map.…”