Conical Kähler–Ricci flows on Fano manifolds

Abstract

In this paper, we study the stability of the conical Kähler-Ricci flows on Fano manifolds. That is, if there exists a conical Kähler-Einstein metric with cone angle 2πβ along the divisor, then for any β ′ sufficiently close to β, the corresponding conical Kähler-Ricci flow converges to a conical Kähler-Einstein metric with cone angle 2πβ ′ along the divisor. Here, we only use the condition that the Log Mabuchi energy is bounded from below. This is a weaker condition than the properness that we have adopted to study the convergence in [36,37]. As corollaries, we give parabolic proofs of Donaldson's openness theorem [17] and his existence conjecture [18] for the conical Kähler-Einstein metrics with positive Ricci curvatures. IntroductionSince the conical Kähler-Einstein metrics play an important role in the solution of the Yau-Tian-Donaldson's conjecture which has been proved by Chen-Donaldson-Sun [6][7][8] and Tian [51], the existence and geometry of the conical Kähler-Einstein metrics have been widely concerned. The conical Kähler-Einstein metrics have been studied by Berman [1], Brendle [3], Campana-Guenancia-Pȃun [4], Donaldson [17], Guenancia-Pȃun [21], Guo-Song [22,23], Jeffres [25], Jeffres-Mazzeo-Rubinstein [26], Li-Sun [32], Mazzeo [38], Song-Wang [48], Tian-Wang [52] and Yao [58] etc. For more details, readers can refer to Rubinstein's article [44].The conical Kähler-Ricci flows were introduced to attack the existence of the conical Kähler-Einstein metrics. These flows were first proposed in Jeffres-Mazzeo-Rubinstein's paper (see Section 2.5 in [26]), then Song-Wang (conjecture 5.2 in [48]) made a conjecture on the relation between the convergence of these flows and the greatest Ricci lower bounds of the manifolds. Then the existence, regularity and limit behavior of the conical Kähler-Ricci flows have been studied by Chen-Wang [9,10], Edwards [19,20], Liu-Zhang [34], Liu-Zhang [36,37], Nomura [39], Shen [45, 46], Wang [57] and Zhang [62, 63] etc.Let M be a Fano manifold with complex dimension n, ω 0 ∈ c 1 (M ) be a smooth Kähler metric and D ∈ | − λK M | (0 < λ ∈ Q) be a smooth divisor.

Metric contraction of the cone divisor by the conical Kähler–Ricci Flow

“…Moreover, π * ω T defines a conical Kähler metric on (P 2 \ {y 0 }, D ′ \ {y 0 }) in the sense that π * ω T is a smooth Kähler metric on Y \ D ′ , and for all x ∈ π(F i ), x = y 0 , π * ω T is quasi-isometric to √ −1 dz 1 ∧ dz 1 |z 1 | 2β i + dz 2 ∧ dz 2 in a coordinate patch U ⊂⊂ P 2 \ {y} centered at x with coordinates (z 1 , z 2 ) such that π(F i ) ∩ U = {z 1 = 0}. (ii) If k = 1, β ∈ (0, 1), and 2a b−a = (1 − β), then the initial Kähler class is a positive multiple of [K −1 X ] − [D], and (X, ω(t)) Gromov-Hausdorff converges to a single point at the singularity time, as proved by Liu-Zhang [24].…”

“…equation (3.2)) and the initial Kähler class is a positive multiple of it. Thus the conical Kähler-Ricci flow will converge in Gromov-Hausdorff topology to a single point at the singularity time as proved by Liu-Zhang [24]. Remark 1.5.…”

“…While the conical Kähler-Ricci flow has been studied for many different geometric situations [11,15,24,27,34,49], little is known about the geometry of finite time singularities of the flow in general. It is reasonable to expect that singularities of the conical Kähler-Ricci flow may behave similarly to those of the Kähler-Ricci flow, however the analysis of singularities along the conical Kähler-Ricci flow is complicated by the presence of the cone divisor.…”

“…We refer to condition (1.2) as the contracting case, and condition (1.3) as the collapsing case. Condition (1.4) corresponds to the log Fano case and is due to Liu-Zhang [24] (cf. Remark 1.4).…”

On a twisted conical Kähler–Ricci flow

“…Proof of Lemma 4.1. We need to use a smooth approximation for the conical equations (1.8) and (2.3), introduced in [17,40] and also used in e.g. [24,6].…”