Étude d'une équation de Monge-Ampère sur les variétés kählériennes compactes

Cherrier, Pascal

doi:10.1016/0022-1236(83)90033-2

Cited by 9 publications

(13 citation statements)

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“…Cherrier [Che87] proved this result by generalizing the elliptic approach of Aubin [Aub78] and [Yau78].…”

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

confidence: 97%

“…It follows from [Che87] that (5.5) admits an unique smooth χ-psh solution, therefore there exists an unique twisted Einstein metric in −(c BC 1 (X) − {η}).…”

Section: Twisted Einstein Metric On Hermitian Manifoldsmentioning

confidence: 99%

“…We now use a result in elliptic Monge-Ampère equation due to Cherrier [Che87] to prove the existence of twisted Einstein metric. An alternative proof using the convergence of the twisted Chern-Ricci flow will be given in Theorem 5.3.…”

Section: Twisted Einstein Metric On Hermitian Manifoldsmentioning

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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“…Cherrier [Che87] proved this result by generalizing the elliptic approach of Aubin [Aub78] and [Yau78].…”

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

confidence: 97%

“…It follows from [Che87] that (5.5) admits an unique smooth χ-psh solution, therefore there exists an unique twisted Einstein metric in −(c BC 1 (X) − {η}).…”

Section: Twisted Einstein Metric On Hermitian Manifoldsmentioning

confidence: 99%

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

Tô

2017

Math. Ann.

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“…When ω is not Kähler, the existence of the solution of the complex Monge-Ampère equation has been studied under some assumptions on ω (see [Cherrier 1987;Guan and Li 2009;Hanani 1996;Tosatti and Weinkove 2010b]). For a general ω, Tosatti and Weinkove [2010a] obtained the key C 0 -estimate.…”

Section: Introductionmentioning

confidence: 99%

“…Locally nonconstant convex functions, affine functions and peakless functions have been investigated on complete Riemannian manifolds and complete noncompact Busemann G-spaces and Alexandrov spaces in various ways. The topology of Riemannian manifolds admitting convex functions was investigated in [Bangert 1978;Greene and Shiohama 1981b;1981a;1987], and that of Busemann G-surfaces in [Innami 1982a;Mashiko 1999b]. It should be noted that convex functions on complete Alexandrov surfaces are not continuous.…”

Section: Introductionmentioning

confidence: 99%

Untitled

2015

Pacific J. Math.

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Denote by J the operator of coefficient stripping. We show that for any free convolution semigroup f t W t 0g with finite variance, applying a single stripping produces semicircular evolution with nonzero initial condition, J OE t D ť ; , where ˇ; is the semicircular distribution with meanˇand variance . For more general freely infinitely divisible distributions , expressions of the form Q t arise from stripping Q t , where f. Q t ; t / W t 0g forms a semigroup under the operation of two-state free convolution. The converse to this statement holds in the algebraic setting. Numerous examples illustrating these constructions are computed. Additional results include the formula for generators of such semigroups.

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Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator

Delanoë

1991

Ann. Inst. H. Poincaré C Anal. Non Linéaire

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L'accès aux archives de la revue « Annales de l'I. H. P., section C » (http://www.elsevier.com/locate/anihpc) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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Étude d'une équation de Monge-Ampère sur les variétés kählériennes compactes

Cited by 9 publications

References 2 publications

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

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

Untitled

Classical solvability in dimension two of the second boundary-value problem associated with the Monge-Ampère operator

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