Related papers: Monge-Amp\`ere equation, hyperk\"ahler structure a…
We consider the Dirichlet problem for the complex Monge--Amp\`ere equation on strongly pseudoconvex K\"ahler manifolds when the right-hand side is decreasing in the solution. Using flow-based arguments, we establish existence of smooth…
We prove the existence and uniqueness of the solutions of some very general type of degenerate complex Monge-Amp\`ere equations. This type of equations is precisely what is needed in order to construct K\"ahler-Einstein metrics over…
We solve the quaternionic Monge-Amp\`ere equation on hyperK\"ahler manifolds. In this way we prove the ansatz for the conjecture raised by Alesker and Verbitsky claiming that this equation should be solvable on any hyperK\"ahler with…
We study asymptotic behaviors of solutions to the Monge-Amp\`ere equation in cones and use the expansion as a tool to study the regularity of solutions in polygonal domains.
Let $X$ be a compact K\"ahler manifold and let $\mu$ be a non-pluripolar measure on $X$. We give a necessary and sufficient condition for $\mu$ so that the complex Monge-Amp\`ere equation (in a K\"ahler class in $X$) having $\mu$ as the…
We obtain a quantitative expansion at infinity of solutions for a kind of Monge-Amp\`ere type equations that origin from mean curvature equations of Lagrangian graph $(x,Du(x))$ and refine the previous study on zero mean curvature equations…
We study the Navier-Stokes and Euler equations of incompressible hydrodynamics in two and three spatial dimensions and show how the constraint of incompressiblility leads to equations of Monge--Amp\`ere type for the stream function, when…
We define a nondegenerate Monge-Amp\`ere structure on a 6-dimensional manifold as a pair $(\Omega,\omega)$, such that $\Omega$ is a symplectic form and $\omega$ is a 3-differential form which satisfies $\omega\wedge\Omega=0$ and which is…
We give a simple interpretation of the adapted complex structure of Lempert-Szoke and Guillemin-Stenzel: it is given by a polar decomposition of the complexified manifold. We then give a twistorial construction of an SO(3)-invariant…
Given a compact K\"ahler manifold, we survey the study of complex Monge-Amp\`ere type equations with prescribed singularity type, developed by the authors in a series of papers. In addition, we give a general answer to a question of…
We prove that a complete noncompact K\"ahler surface with positive and bounded sectional curvature is biholomorphic to $\mathbb{C}^2$. This result confirms a special case of Yau's conjecture that a complete noncompact K\"ahler $n$-manifold…
On a K\"ahler manifold we consider the problems of maximizing/minimizing Monge--Amp\`ere energy over certain subsets of the space of K\"ahler potentials. Under suitable assumptions we prove that solutions to these variational problems…
In this paper, we study the ellipticity of the vector bundle versions of the Monge-Amp\`ere, $J$, dHYM and $\sigma_{k}$-equations at a point. These are nonlinear geometric partial differential equations defined on a holomorphic vector…
On a compact K\"ahler manifold $(X,\omega)$, we study the strong continuity of solutions with prescribed singularities of complex Monge-Amp\`ere equations with integrable Lebesgue densities. Moreover, we give sufficient conditions for the…
We investigate the Monge-Amp\`ere equation subject to zero boundary value and with a positive right-hand side unction assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are…
We introduce a new approach to Monge-Ampere geometry based on techniques from higher symplectic geometry. Our work is motivated by the application of Monge-Ampere geometry to the Poisson equation for the pressure that arises for…
A notion of asymptotically conical K\"ahler orbifold is introduced, and, following previous existence results in the setting of asymptotically conical manifolds, it is shown that a certain complex Monge-Amp\'ere equation admits a rapidly…
It is proved that solutions of the complex Monge-Amp\`ere equation on compact K\"ahler manifolds with right hand side in $L^p, p>1$ are uniformly H\"older continuous under the assumption on non-negative orthogonal bisectional curvature.
Uniform bounds are obtained using the auxiliary Monge-Amp\`ere equation method for solutions of very general classes of fully non-linear partial differential equations, assuming the existence of a ${C}$-subsolution in the sense of G.…
Let X be a smooth projective Berkovich space over a complete discrete valuation field K of residue characteristic zero, and assume that X is defined over a function field admitting K as a completion. Let further m be a positive measure on X…