On a twisted conical Kähler–Ricci flow

Zhang, Yashan

doi:10.1007/s10455-018-9619-z

Cited by 3 publications

(1 citation statement)

References 52 publications

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Conical Kähler–Ricci flows on Fano manifolds

Liu

Zhang

2017

Advances in Mathematics

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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.

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