Complex Hessian Equation on Kähler Manifold

Hou, Zuoliang

doi:10.1093/imrn/rnp043

Cited by 47 publications

(32 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Similar results were obtained by Hou [13] by obtaining a Laplacian estimate for complex Hessian equations depending on inf M Δf 1 m in the case of non-negative orthogonal bisectional curvature. The main difficulty in improving the exponent from 1/m to 1/(m − 1) is that we can no longer discard terms by using the concavity of (det B)…”

Section: Introductionsupporting

confidence: 84%

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

Picard

2013

Mathematical Research Letters

View full text Add to dashboard Cite

Abstract. The regularity theory of the degenerate complex Monge-Ampère equation is studied. The equation is considered on a closed compact Kähler manifold (M, g) with non-negative orthogonal bisectional curvature of dimension m. Given a solution ϕ of the degenerate complex Monge-Ampère equation det(g ij + ϕ ij ) = f det(g ij ), it is shown that the Laplacian of ϕ can be controlled by a constant depending on (M, g), sup f , and inf M Δf 1/(m−1) .

show abstract

Section: Introductionsupporting

confidence: 84%

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

Picard

2013

Mathematical Research Letters

View full text Add to dashboard Cite

show abstract

“…Among them, one important example is Yau's seminal work on the complex Monge-Ampère equations in the Calabi conjecture. The general case has been studied recently in [Hou 2009;Hou et al 2010]. There exist, however, some analytical difficulties in completely solving this problem for k < n.…”

Section: Introductionmentioning

confidence: 99%

On the geometric flows solving Kählerian inverseσ_kequations

Fang¹,

Lai²

2012

Pacific J. Math.

View full text Add to dashboard Cite

show abstract

“…This result was known to hold when (X, ω) has non negative holomorphic bisectional curvature [H09,Jb10].…”

Section: Preliminariesmentioning

confidence: 82%

“…The non degenerate complex Hessian equation on compact Kähler manifold, where F (x, ϕ) = f (x), with 0 < f ∈ C ∞ (X), has been studied recently in [H09,HMW10,Jb10,DK12]. In [H09] and [Jb10], the authors independently solved this equation with a strong additional hypothesis, assuming (X, ω) has non negative holomorphic bisectional curvature.…”

Section: Introductionmentioning

confidence: 99%

Solutions to degenerate complex Hessian equations

2013

Journal de Mathématiques Pures et Appliquées

View full text Add to dashboard Cite

Abstract. Let (X, ω) be an n-dimensional compact Kähler manifold. We study degenerate complex Hessian equations of the form (ω+dd c ϕ) m ∧ω n−m = F (x, ϕ)ω n . Under some natural conditions on F , this equation has a unique continuous solution. When (X, ω) is rational homogeneous we further show that the solution is Hölder continuous. IntroductionLet (X, ω) be a compact Kähler manifold of complex dimension n. Fix an integer m between 1 and n, and let d, dc denote the usual real differentialWe are studying degenerate complex Hessian equations of the formwhere the density F : X × R → R + satisfies some natural integrability conditions (see Theorem A below).The case m = 1 corresponds to the Laplace equation and the case m = n corresponds to degenerate complex Monge-Ampère equations which have been studied intensively in recent years (see [Bl03, Bl05, Bl12, BGZ08, BK07, EGZ09, GKZ08, GZ05, GZ07, Kol98, Kol02, Kol03, Kol05]). So, equation ( Following Blocki [Bl05] we develop a potential theory for the complex Hessian equation on compact Kähler manifold. We define the class of (ω, m)-subharmonic functions which is a generalization of the class of ω-plurisubharmonic functions when m = n. The definition of the complex Hessian operator on bounded (ω, m)-subharmonic functions is delicate due to difficulties in regularization process.To go around this difficulty, we introduce a capacity and use it to define the concept of quasi-uniform convergence. This allows us to define a suitable class of bounded and quasi-continuous (ω, m)-subharmonic functions on which the complex Hessian operator is well defined and continuous under quasi-uniform convergence.Date: July 13, 2017.

show abstract

Complex Hessian Equation on Kähler Manifold

Cited by 47 publications

References 16 publications

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

On the geometric flows solving Kählerian inverseσ_kequations

Solutions to degenerate complex Hessian equations

Contact Info

Product

Resources

About

Complex Hessian Equation on Kähler Manifold

Cited by 47 publications

References 16 publications

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

Priori estimates of the degenerate Monge-Ampère equation on Kähler manifolds of non-negative bisectional curvature

On the geometric flows solving Kählerian inverseσkequations

Solutions to degenerate complex Hessian equations

Contact Info

Product

Resources

About

On the geometric flows solving Kählerian inverseσ_kequations