### A characterization of bounded plurisubharmonic functions

We give a characterization for boundedness of plurisubharmonic functions in the Cegrell class ℱ.

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We give a characterization for boundedness of plurisubharmonic functions in the Cegrell class ℱ.

We modify an example due to X.-J. Wang and obtain some counterexamples to the regularity of the degenerate complex Monge-Ampère equation on a ball in ℂⁿ and on the projective space ℙⁿ.

We prove a decomposition theorem for complex Monge-Ampère measures of plurisubharmonic functions in connection with their pluripolar sets.

We study a general Dirichlet problem for the complex Monge-Ampère operator, with maximal plurisubharmonic functions as boundary data.

For a regular, compact, polynomially convex circled set $K$ in ${\mathbf{C}}^{2}$, we construct a sequence of pairs $\{{P}_{n},{Q}_{n}\}$ of homogeneous polynomials in two variables with $\mathrm{deg}\phantom{\rule{0.166667em}{0ex}}{P}_{n}=$$\mathrm{deg}\phantom{\rule{0.166667em}{0ex}}{Q}_{n}...$

We prove the almost ${}^{1,1}$ regularity of the degenerate complex Monge-Ampère equation in a special case.

Let $X$ be a compact Kähler manifold and $\omega $ be a smooth closed form of bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal{E}}_{\chi}(X,\omega )$ of $\omega $-plurisubharmonic functions of finite weighted Monge-Ampère energy. When the weight $\chi $ 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 ${\mathcal{E}}_{\chi}(X,\omega )$. This is done by establishing...

${C}^{1,1}$ regularity of the solutions of the complex Monge-Ampère equation in ℂPⁿ with the n-root of the right hand side in ${C}^{1,1}$ is proved.

We prove a comparison principle for the log canonical threshold of plurisubharmonic functions under an assumption on complex Monge-Ampère measures.

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...

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

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.

We show that in the class of compact, piecewise ${C}^{1}$ curves K in ${\mathbb{R}}^{n}$, the semialgebraic curves are exactly those which admit a Bernstein (or a van der Corput-Schaake) type inequality for the derivatives of (the traces of) polynomials on K.

We study boundary values of functions in Cegrell’s class ${}_{\psi}$.