Related papers: Monge-Amp\`ere equation, hyperk\"ahler structure a…
Let $(X,\omega)$ be a compact $n$-dimensional K\"ahler manifold on which the integral of $\omega^n$ is $1$. Let $K$ be an immersed real $\mathcal{C}^3$ submanifold of $X$ such that the tangent space at any point of $K$ is not contained in…
In this paper, we study weak solutions to complex Monge-Amp\`ere equations of the form $(\omega + dd^c \varphi)^n= F(\varphi,.)d\mu$ on a bounded strictly pseudoconvex domain in $\mathbb{C}^n$, where $\omega$ is a smooth $(1,1)$-form,…
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 how a symmetry reduction of the equations for incompressible hydrodynamics in three dimensions leads naturally to a Monge-Amp\`ere structure, and Burgers'-type vortices are a canonical class of solutions associated with this…
We prove the existence of unique smooth solutions to the quaternionic Monge-Amp\`{e}re equation for $(n-1)$-quaternionic plurisubharmonic functions on a hyperK\"{a}hler manifold and thus obtain solutions for the quaternionic form type…
Let $X$ be a compact complex manifold which admits a hermitian metric satisfying a curvature condition introduced by Guan-Li. Given a semipositive form $\theta$ with positive volume, we define the Monge-Amp\`ere operator for unbounded…
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}…
The goal of this paper is to show that there are strong relations between certain Monge-Amp\`ere integrals appearing in holomorphic Morse inequalities, and asymptotic cohomology estimates for tensor powers of holomorphic line bundles.…
We show that the metric defined by the solution to the tropical Monge-Amp\`ere equation, as defined by Hultgren, Mazzon, and the first two authors, on the boundary of the 3-simplex is asymptotic to the Gross-Wilson metric on $S^2$ near each…
We describe the behavior of the singularities of solutions to degenerate complex Monge-Amp`ere equations on K\"ahler manifolds. This was not resolved since the fundamental paper of S-T Yau \cite{y} on this subject.
We study a parabolic complex Monge-Amp\`{e}re type equation of the form \eqref{MA} on a complete noncompact \K manifold. We prove a short time existence result and obtain basic estimates. Applying these results, we prove that under certain…
This paper is a natural companion of [Alekseevsky D.V., Alonso Blanco R., Manno G., Pugliese F., Ann. Inst. Fourier (Grenoble) 62 (2012), 497-524, arXiv:1003.5177], generalising its perspectives and results to the context of third-order…
Monge-Amp\`ere equations of the form, $u_{xx}u_{yy}-u_{xy}^2=F(u,u_x,u_y)$ arise in many areas of fluid and solid mechanics. Here it is shown that in the special case $F=u_y^4f(u, u_x/u_y)$, where $f$ denotes an arbitrary function, the…
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 study the complex Monge-Amp\` ere operator on compact K\"ahler manifolds. We give a complete description of its range on the set of $\omega-$plurisubharmonic functions with $L^2$ gradient and finite self energy, generalizing to this…
The complex Monge-Amp\`ere equation admits covariant bi-symplectic structure for complex dimension 3, or higher. The first symplectic 2-form is obtained from a new variational formulation of complex Monge- Amp\`ere equation in the framework…
We give necessary and sufficient conditions for existence of solutions to a general system of complex Monge-Amp\`ere equations on Fano horosymmetric manifolds. In particular, we get necessary and sufficient conditions for existence of…
A general solution to the Complex Monge-Amp\`ere equation in a space of arbitrary dimensions is constructed.
Let $L$ be a big and semipositive line bundle on a complex projective manifold $X$, and let $\theta\in c_1(L)$ be a smooth semipositive representative. In the adjoint setting $H^0(X,L^k\otimes K_X)$, we prove that Donaldson's quantized…
In this paper we are concerned with the problem of local and global subextensions of (quasi-)plurisubharmonic functions from a "regular" subdomain of a compact K\"ahler manifold. We prove that a precise bound on the complex Monge-Amp\`ere…