Regularizing properties of complex Monge–Ampère flows

Tô, Tat Dat

doi:10.1016/j.jfa.2016.10.017

Cited by 5 publications

(9 citation statements)

References 18 publications

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“…By generalizing a result of Székelyhidi-Tosatti [SzTo11], Nie [Nie14] has proved this property for compact Hermitian manifolds of vanishing first Bott-Chern class and continous initial data. In this paper, we generalize the previous results of Nie [Nie14] and the author [Tô16] by considering the following complex Monge-Ampère flow:…”

Section: Introductionmentioning

confidence: 59%

“…In Section 1, we recall some notations in Hermitian manifolds. In Section 2 we prove various a priori estimates following our previous work [Tô16]. The main difference is that we will use the recent result of Kołoziedj's uniform type estimates for Monge-Ampère on Hermitian manifolds (cf.…”

Section: Theorem a Let ϕ 0 Be A ω-Psh Function With Zero Lelong Numbmentioning

confidence: 99%

“…Bounding the gradient of ϕ. In this section we bound the gradient of ϕ using the same technique as in [Tô16] (see also [SzTo11]) which is a parabolic version of Błocki's estimate [Bł09] for Kähler manifolds. In these articles we used the usual normal coordinates in Kähler geometry.…”

Section: 2mentioning

confidence: 99%

“…The estimate on |∇ϕ| 2 ω is needed here. The argument follows from [GZ13,Tô16] but there are difficulties in using this approach because of torsion terms that need to be controlled (see also [TW10] for similar computation for the elliptic Monge-Ampère equation).…”

Section: We Now Assume That K(t Z) Attains Its Maximum At (Tmentioning

confidence: 99%

“…Recently, we extended these latter works to deal with general complex MongeAmpère flows and arbitrary initial condition (cf. [Tô16]). One of the motivations for this problem comes from the Analytic Minimal Model Program proposed by Song-Tian [ST16].…”

Section: Introductionmentioning

confidence: 99%

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Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds

Tô

2017

Math. Ann.

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Abstract. We prove that a general complex Monge-Ampère flow on a Hermitian manifold can be run from an arbitrary initial condition with zero Lelong number at all points. Using this property, we confirm a conjecture of Tosatti-Weinkove: the ChernRicci flow performs a canonical surgical contraction. Finally, we study a generalization of the Chern-Ricci flow on compact Hermitian manifolds, namely the twisted ChernRicci flow. IntroductionLet (X, g, J) be a compact Hermitian manifold of complex dimension n, that is a compact complex manifold such that J is compatible with the Riemannian metric g. Recently a number of geometric flows have been introduced to study the structure of Hermitian manifolds. Some flows which do preserve the Hermitian property have been proposed by Streets-Tian [ST10, ST11, ST13], Liu-Yang [LY12] and also anomaly flows due to Phong-Picard-Zhang [PPZ15, PPZ16a, PPZ16b] which moreover preserve the conformally balanced condition of Hermitian metrics. Another such flow, namely the Chern-Ricci flow, was introduced by Gill [Gil11] and has been further developed by Tosatti-Weinkove in [TW15]. The Chern-Ricci flow is written aswhere Ric(ω) is the Chern-Ricci form which is defined locally by Assume that there exists a holomorphic map between compact Hermitian manifolds π : X → Y blowing down an exceptional divisor E on X to one point y 0 ∈ Y . In addition, assume that there exists a smooth function ρ on X such thatwith T < +∞. Tosatti and Weinkove proved:Observe that ω T induces a singular metric ω ′ on Y which is smooth in Y \ {y 0 }. In this note, we confirm this conjecture 1 . An essential ingredient of its proof is to prove that the Monge-Ampère flow corresponding to the Chern-Ricci flow can be run from a rough data. By generalizing a result of Székelyhidi-Tosatti [SzTo11], Nie [Nie14] has proved this property for compact Hermitian manifolds of vanishing first Bott-Chern class and continous initial data. In this paper, we generalize the previous results of Nie [Nie14] and the author [Tô16] by considering the following complex Monge-Ampère flow:where (θ t ) t∈[0,T ] is a family of Hermitian forms with θ 0 = ω and F is a smooth function on R × X × R. Theorem A. Let ϕ 0 be a ω-psh function with zero Lelong number at all points. Let (t, z, s) → F (t, z, s) be a smooth function on [0, T ] × X × R such that ∂F/∂s ≥ 0 and ∂F/∂t is bounded from below. Then there exists a family of smooth strictly is continuous. This family is moreover unique if ∂F/∂t is bounded and ∂F/∂s ≥ 0.The following stability result is a straighforward extension of [Tô16, Theorem 4.3, 4.4].1 After this paper was completed, the author learned that Xiaolan Nie proved the first statement of the conjecture for complex surfaces (cf. [Nie17]). She also proved that the Chern-Ricci flow can be run from a bounded data. The author would like to thank Xiaolan Nie for sending her preprint. REGULARIZING PROPERTIES OF COMPLEX MONGE-AMPÈRE FLOWS II 3Theorem B. Let ϕ 0 , ϕ 0,j be ω-psh functions with zero Lelong number at all points, such that ϕ 0,...

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Section: Introductionmentioning

confidence: 59%

Section: Theorem a Let ϕ 0 Be A ω-Psh Function With Zero Lelong Numbmentioning

confidence: 99%

Section: 2mentioning

confidence: 99%

Section: We Now Assume That K(t Z) Attains Its Maximum At (Tmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds

Tô

2017

Math. Ann.

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Pluripotential Kähler–Ricci flows

Guedj

Zériahi

2020

Geom. Topol.

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The complex Hessian quotient flow on compact Hermitian manifolds

Zhou¹,

Chu

2022

MATH

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<abstract><p>In this paper, we consider the parabolic Hessian quotient equation on compact Hermitian manifolds. By setting up a priori estimates of the admissible solutions, we prove the long-time existence of the solution to the parabolic Hessian quotient equation and its convergence. As an application, we show the solvability of a class of complex Hessian quotient equations, which generalizes the relevant results.</p></abstract>

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Regularizing properties of complex Monge–Ampère flows

Cited by 5 publications

References 18 publications

Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds

Regularizing properties of complex Monge–Ampère flows II: Hermitian manifolds

Pluripotential Kähler–Ricci flows

The complex Hessian quotient flow on compact Hermitian manifolds

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