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A Hilbert Lemniscate Theorem in 2

Thomas Bloom, Norman Levenberg, Yu. Lyubarskii (2008)

Annales de l’institut Fourier

For a regular, compact, polynomially convex circled set K in C 2 , we construct a sequence of pairs { P n , Q n } of homogeneous polynomials in two variables with deg P n = deg Q n ...

A priori estimates for weak solutions of complex Monge-Ampère equations

Slimane Benelkourchi, Vincent Guedj, Ahmed Zeriahi (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Let X be a compact Kähler manifold and ω be a smooth closed form of bidegree ( 1 , 1 ) which is nonnegative and big. We study the classes χ ( X , ω ) of ω -plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight χ has fast growth at infinity, the corresponding functions are close to be bounded. We show that if a positive Radon measure is suitably dominated by the Monge-Ampère capacity, then it belongs to the range of the Monge-Ampère operator on some class χ ( X , ω ) . This is done by establishing...

A viscosity approach to degenerate complex Monge-Ampère equations

Ahmed Zeriahi (2013)

Annales de la faculté des sciences de Toulouse Mathématiques

This is the content of the lectures given by the author at the winter school KAWA3 held at the University of Barcelona in 2012 from January 30 to February 3. The main goal was to give an account of viscosity techniques and to apply them to degenerate Complex Monge-Ampère equations.We will survey the main techniques used in the viscosity approach and show how to adapt them to degenerate complex Monge-Ampère equations. The heart of the matter in this approach is the “Comparison Principle" which allows...

About the Calabi problem: a finite-dimensional approach

H.-D. Cao, J. Keller (2013)

Journal of the European Mathematical Society

Let us consider a projective manifold M n and a smooth volume form Ω on M . We define the gradient flow associated to the problem of Ω -balanced metrics in the quantum formalism, the Ω -balancing flow. At the limit of the quantization, we prove that (see Theorem 1) the Ω -balancing flow converges towards a natural flow in Kähler geometry, the Ω -Kähler flow. We also prove the long time existence of the Ω -Kähler flow and its convergence towards Yau’s solution to the Calabi conjecture of prescribing the...

An energy estimate for the complex Monge-Ampère operator

Urban Cegrell, Leif Persson (1997)

Annales Polonici Mathematici

We prove an energy estimate for the complex Monge-Ampère operator, and a comparison theorem for the corresponding capacity and energy. The results are pluricomplex counterparts to results in classical potential theory.

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