Expansion formula for complex Monge–Ampère equation along cone singularities

Yin, Hao; Zheng, Kai

doi:10.1007/s00526-019-1498-z

Cited by 13 publications

(16 citation statements)

References 15 publications

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“…Remark For any cone angle

0 < β < 1

, general expansion formulas for Kähler–Einstein cone metrics appear in . They come from the study of a singular Monge–Ampère equation.…”

Section: Csck Metrics With Cone Singularitiesmentioning

confidence: 99%

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Construction of constant scalar curvature Kähler cone metrics

Keller

Zheng

2018

Proc. London Math. Soc.

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Abstract. Over a compact Kähler manifold, we provide a Fredholm alternative result for the Lichnerowicz operator associated to a Kähler metric with conic singularities along a divisor. We deduce several existence results of constant scalar curvature Kähler metrics with conic singularities: existence result under small deformations of Kähler classes, existence result over a Fano manifold, existence result over certain ruled manifolds. In this last case, we consider the projectivisation of a parabolic stable holomorphic bundle. This leads us to prove that the existing Hermitian-Einstein metric on this bundle enjoys a regularity property along the divisor on the base.

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“…Remark For any cone angle

0 < β < 1

, general expansion formulas for Kähler–Einstein cone metrics appear in . They come from the study of a singular Monge–Ampère equation.…”

Section: Csck Metrics With Cone Singularitiesmentioning

confidence: 99%

“…But we consider here the theorem only for angle

β < min (1 / 2, δ_{0})

. In general, we believe the asymptotic behavior of Hermitian–Einstein metric

h_{E}

could be well understood with the method in and our angle restriction could be removed.…”

Section: Hermitian–einstein Metrics With Conical Singularitiesmentioning

confidence: 99%

Construction of constant scalar curvature Kähler cone metrics

Keller

Zheng

2018

Proc. London Math. Soc.

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“…is invertible. Hence (25) has a solution of the type of (27). When σ + n + m = 0, we need to enlarge the solution space to include…”

Section: Functions In Span(t ) and Span( T ) The Discussion In This mentioning

confidence: 99%

“…For an example in [13] the authors studied Kahler-Einstein metrics with edge singularities and got an expansion for the potential function of the metric along the given divisor, in which the power of log ρ is controlled by a positive integer depending on the power of ρ in every single term, where ρ is the distance to the divisor. Later in [27] this dependence was shown to be rather simple.…”

Section: Introductionmentioning

confidence: 94%

On Expansions of Ricci Flat ALE Metrics in Harmonic Coordinates About the Infinity

Chen

2019

Commun. Math. Stat.

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“…Higher order estimates. We use the same terminology from [54]. We consider asymptotic for cscK cone metrics in the unit ball B 1 ⊂ B * := C × C n−1 centred at the origin,…”

Section: 3mentioning

confidence: 99%

Geodesics in the Space of Kähler Cone Metrics II: Uniqueness of Constant Scalar Curvature Kähler Cone Metrics

Zheng

2019

Comm Pure Appl Math

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In this article, we give a complete construction of geodesics in the space of Kähler cone metrics (cone geodesics), and we address the problem on the uniqueness of constant scalar curvature Kähler (cscK) cone metrics when the cone angle β stays in the whole interval (0, 1]. The part β∈[)12,1 requires new weighted function spaces and new analytic techniques. We determine the asymptotic behavior of both cone geodesics and cscK cone metrics, prove the reductivity of the automorphism group, and establish the linear theory for the Lichnerowicz operator, which immediately implies the openness of the path deforming the cone angles of cscK cone metrics. © 2019 Wiley Periodicals, Inc.

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Expansion formula for complex Monge–Ampère equation along cone singularities

Abstract: In this paper, we prove the asymptotic expansion of the solutions to some singular complex Monge-Ampère equation which arise naturally in the study of the conical Kähler-Einstein metric.

Cited by 13 publications

References 15 publications

Construction of constant scalar curvature Kähler cone metrics

Construction of constant scalar curvature Kähler cone metrics

On Expansions of Ricci Flat ALE Metrics in Harmonic Coordinates About the Infinity

Geodesics in the Space of Kähler Cone Metrics II: Uniqueness of Constant Scalar Curvature Kähler Cone Metrics

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