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Hölder continuous solutions to Monge–Ampère equations

Jean-Pierre Demailly, Sławomir Dinew, Vincent Guedj, Pham Hoang Hiep, Sławomir Kołodziej, Ahmed Zeriahi (2014)

Journal of the European Mathematical Society

Let ( X , ω ) be a compact Kähler manifold. We obtain uniform Hölder regularity for solutions to the complex Monge-Ampère equation on X with L p right hand side, p > 1 . The same regularity is furthermore proved on the ample locus in any big cohomology class. We also study the range ( X , ω ) of the complex Monge-Ampère operator acting on ω -plurisubharmonic Hölder continuous functions. We show that this set is convex, by sharpening Kołodziej’s result that measures with L p -density belong to ( X , ω ) and proving that ( X , ω ) has the...

Hölder regularity for solutions to complex Monge-Ampère equations

Mohamad Charabati (2015)

Annales Polonici Mathematici

We consider the Dirichlet problem for the complex Monge-Ampère equation in a bounded strongly hyperconvex Lipschitz domain in ℂⁿ. We first give a sharp estimate on the modulus of continuity of the solution when the boundary data is continuous and the right hand side has a continuous density. Then we consider the case when the boundary value function is 1 , 1 and the right hand side has a density in L p ( Ω ) for some p > 1, and prove the Hölder continuity of the solution.

Infinite geodesic rays in the space of Kähler potentials

Claudio Arezzo, Gang Tian (2003)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we prove the existence of solutions of a degenerate complex Monge-Ampére equation on a complex manifold. Applying our existence result to a special degeneration of complex structure, we show how to associate to a change of complex structure an infinite length geodetic ray in the space of potentials. We also prove an existence result for the initial value problem for geodesics. We end this paper with a discussion of a list of open problems indicating how to relate our reults to the...

Jensen measures, hyperconvexity and boundary behaviour of the pluricomplex Green function

Magnus Carlehed, Urban Cegrell, Frank Wikström (1999)

Annales Polonici Mathematici

We characterise hyperconvexity in terms of Jensen measures with barycentre at a boundary point. We also give an explicit formula for the pluricomplex Green function in the Hartogs triangle. Finally, we study the behaviour of the pluricomplex Green function g(z,w) as the pole w tends to a boundary point.

Kähler-Einstein metrics with mixed Poincaré and cone singularities along a normal crossing divisor

Henri Guenancia (2014)

Annales de l’institut Fourier

Let X be a compact Kähler manifold and Δ be a ℝ -divisor with simple normal crossing support and coefficients between 1 / 2 and 1 . Assuming that K X + Δ is ample, we prove the existence and uniqueness of a negatively curved Kahler-Einstein metric on X ∖ Supp ( Δ ) having mixed Poincaré and cone singularities according to the coefficients of Δ . As an application we prove a vanishing theorem for certain holomorphic tensor fields attached to the pair ( X , Δ ) .

L1 and L∞-estimates with a local weight for the ∂-equation on convex domains in Cn.

Francesc Tugores (1992)

Publicacions Matemàtiques

We construct a defining function for a convex domain in Cn that we use to prove that the solution-operator of Henkin-Romanov for the ∂-equation is bounded in L1 and L∞-norms with a weight that reflects not only how near the point is to the boundary of the domain but also how convex the domain is near the point. We refine and localize the weights that Polking uses in [Po] for the same type of domains because they depend only on the Euclidean distance to the boudary and don't take into account the...

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