Related papers: The Monge-Amp\`{e}re equation for $(n-1)$-quaterni…
We study a fully nonlinear equation of complex Monge-Ampere type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.
In this paper, we prove a uniform and sharp estimate for the modulus of continuity of solutions to complex Monge-Amp\`ere equations, using the PDE-based approach developed by the first three authors in their approach to supremum estimates…
We prove the existence and uniqueness of continuous solutions to the complex Monge-Amp\`ere type equation with the right hand side in $L^p$, $p>1$, on compact Hermitian manifolds. Next, we generalise results of Eyssidieux, Guedj and Zeriahi…
In this paper, we give some precise characterizations of existence of solution to the complex Monge - Amp\`ere equation in the classes $\mathcal E_\chi(\Omega)$ and $\mathcal E_{\chi,loc}(\Omega)$.
In this paper we continue the analysis of the two-scale method for the Monge-Amp\`ere equation for dimension $d \geq 2$ introduced in [10]. We prove continuous dependence of discrete solutions on data that in turn hinges on a discrete…
We prove that on a compact K\"ahler manifold, the non-pluripolar Monge-Amp\`ere mass of a $\theta$-psh function decreases as the singularities increase. This was conjectured by Boucksom-Eyssidieux-Guedj-Zeriahi who proved it under the…
We propose an extension to our monotone and convergent method for the Monge-Amp\`{e}re equation in dimension $d \geq2$, that incorporates the idea of filtered schemes. The method combines our original monotone operator with a more accurate…
We introduce generalized Monge-Amp\`ere capacities and use these to study complex Monge-Amp\`ere equations whose right-hand side is smooth outside a divisor. We prove, in many cases, that there exists a unique normalized solution which is…
We give a new proof for the interior regularity of strictly convex solutions of the Monge-Amp\`ere equation. Our approach uses a doubling inequality for the Hessian in terms of the extrinsic distance function on the maximal Lagrangian…
We prove a regularity result for the Monge--Amp\`ere equations on compact Kaehler manifolds with degenerate rhs member.
Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds.…
Let $(X,\omega)$ be a compact K\"ahler manifold. We prove the existence and uniqueness of solutions to complex Monge-Amp\`ere equations with prescribed singularity type. Compared to previous work, the assumption of small unbounded locus is…
We consider smooth solutions to the Monge-Amp`ere equation subject to mixed boundary conditions on annular domains. We establish global $C^2$ estimates when the boundary of the domain consists of two smooth strictly convex closed…
We introduce a wide subclass ${\cal F}(X,\omega)$ of quasi-plurisubharmonic functions in a compact K\"ahler manifold, on which the complex Monge-Amp\`ere operator is well-defined and the convergence theorem is valid. We also prove that…
Finite energy pluripotential theory accommodates the variational theory of equations of complex Monge-Amp\`ere type arising in K\"ahler geometry. Recently it has been discovered that many of the potential spaces involved have a rich metric…
We study the Dirichlet problem for the complex Monge-Amp\`ere operator on a B-regular domain $\Omega$, allowing boundary data that is singular or unbounded. We introduce the concept of pluri-quasibounded functions on $\Omega$ and $\partial…
The main goal of this article is to find, following the approach given in [Ce1] and [Ce2], the largest possible sub-class of plurisubharmornic functions on a complex variety on which the complex Monge-Amp\`ere operator can be reasonably…
The aim of this article is to study the residual Monge-Amp\`{e}re mass of a plurisubharmonic function with an isolated singularity, provided with the circular symmetry. With the aid of Sasakian geometry, we obtain an estimate on the…
In this paper, we study the convergence in the capacity of sequence of plurisubharmonic functions. As an application, we prove stability results for solutions of the complex Monge-Amp\`ere equations.
We study various capacities on compact K\"{a}hler manifolds which generalize the Bedford-Taylor Monge-Amp\`ere capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Amp\`ere…