Related papers: The General Solution of the Complex Monge-Amp\`ere…

A general solution to the Complex Monge-Amp\`ere equation in a space of arbitrary dimensions is constructed.

It is shown that the general solution of a homogeneous Monge-Amp\`{e}re equation in $n$-dimensional space is closely connected with the exactly (but only implicitly) integrable system \frac {\partial \xi_{j}}{\partial x_0}+\sum_{k=1}^{n-1}…

A general solution to the Complex Bateman equation in a space of arbitrary dimensions is constructed.

We study generalized complex Monge-Amp\`ere type equations on closed Hermitian manifolds. We derive {\em a priori} estimates and then prove the existence of admissible solutions. Moreover, the gradient estimate is improved.

We construct several types of multi-valued solutions to the Monge-Ampere equation in higher dimensions.

We associate an integrable generalized complex structure to each 2-dimensional symplectic Monge-Amp\`ere equation of divergent type and, using the Gualtieri $\bar{\partial}$ operator, we characterize the conservation laws and the generating…

We show existence and uniqueness of solutions to the Monge-Ampere equation on compact almost complex manifolds with non-integrable almost complex structure.

In this short note, we prove the existence of solutions to a Monge-Amp\`ere equation of entire type derived by a weighted version of the classical Minkowski problem.

We show a very general existence theorem to the complex Monge-Amp\`ere type equation on hyperconvex domains.

We obtain a genuine local $C^2$ estimate for the Monge-Amp\`ere equation in dimension two, by using the partial Legendre transform.

We consider the complex Monge-Amp\`ere equation with an additional linear gradient term inside the determinant. We prove existence and uniqueness of solutions to this equation on compact Hermitian manifolds.

We consider a generalised complex Monge-Amp\`ere equation on a compact K\"ahler manifold and treat it using the method of continuity. For complex surfaces, we prove an easy existence result. We also prove that (for three-folds and a related…

We show a general existence theorem to the complex Monge-Amp\`ere type equation on compact K\"ahler manifolds.

In this paper we consider the generalised solutions to the Monge-Amp{\`{e}}re type equations with general source terms. We firstly prove the so-called comparison principle and then give some important propositions for the border of…

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 $\mathcal E_{\chi,loc}(\Omega)$.

We show that, up to scaling, the complex Monge-Ampere equation on compact Hermitian manifolds always admits a smooth solution.

We construct solutions to Monge-Amp\`ere equations whose Monge-Amp\`ere measures contain singular components supported on low codimensional sets. We also study the regularity of such solutions. To motivate our construction, we present…

In this paper, we obtain gradient estimates and Laplacian estimates for the solution to the singular complex Monge-Amp\`ere equation by applying the integral method.

We study the Dirichlet problem for the Monge-Amp\`ere equation on almost complex manifolds. We obtain the existence of the unique smooth solution of this problem in strictly pseudoconvex domains.

We consider the complex Monge-Amp\'{e}re equation on complete K\"{a}hler manifolds with cusp singularity along a divisor when the right hand side $F$ has rather weak regularity. We proved that when the right hand side $F$ is in some…