Equations du type de Monge–Ampère sur les variétés hermitiennes compactes

Hanani, Abdellah

doi:10.1006/jfan.1996.0040

Cited by 52 publications

(45 citation statements)

References 2 publications

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“…We also refer the reader to [Ha96, Lemma 6] for a gradient estimate for the elliptic Complex Mong-Ampère equation in the Hermitian case without using the local coordinate system. We thank Valentino Tosatti for indicating the reference [Ha96]. We remark that similar arguments of the proof below can be found in [Nie14, Lemma 3.3].…”

Section: 2supporting

confidence: 52%

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: 2supporting

confidence: 52%

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

Tô

2017

Math. Ann.

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“…In the smooth category, for λ > 0, given f positive and smooth Cherrier [Ch87] proved the existence and uniqueness of the smooth admissible solution. Later on, more general results are proved in this case by Hanani [Ha96a,Ha96b].…”

Section: Introductionmentioning

confidence: 87%

The complex Monge–Ampère type equation on compact Hermitian manifolds and applications

Nguyen

2016

Advances in Mathematics

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“…Signalons au passage que la fonctionnelle conside re e dans la de monstration du Lemme 1 de [9], aÁ savoir 1(. )= |{.| 2 exp[e &k. ], k>0, ne permet pas de conclure; les calculs propose s ne cessitent une majoration a priori de .…”

Section: De Finitions Et Notationsunclassified

“…# B. En proce dant, comme au Lemme 8 dans [8], aÁ des e quivalents de l'ordre de (1+%)(1+ 2 ) preÁ s, on montre (4 …”

Section: De Finitions Et Notationsunclassified

“…Ce travail constitue la suite d'une e tude portant sur des e quations aux de rive es partielles elliptiques non line aires du type de Monge AmpeÁ re sur des varie te s hermitiennes [8 10]; il s'agit d'une extension aux varie te s aÁ bord non vide de re sultats de [9]. On consideÁ re une varie te C hermitienne aÁ bord strictement pseudoconvexe, et on s'inte resse aÁ la mise en e vidence de solutions des probleÁ mes de Dirichlet suivants: dans X sur X (0.1) et { det($ En vue d'e tablir l'existence d'hypersurfaces de R m+1 aÁ courbure de Gauss prescrite, on retrouve l'analogue re el de ces probleÁ mes; cette question a e te re solue par Oliker [13], Delanoe , Caffarelli, Nirenberg, Spruck [5] .…”

Section: Introductionunclassified

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