Related papers: Modulus of continuity for solutions to complex Mon…
We prove existence and regularity of entire solutions to Monge-Ampere equations invariant under an irreducible action of a compact Lie group.
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…
We study the first derivative estimates for solutions to Monge-Amp\`ere equations in terms of modulus of continuity. As a result, we establish the optimal global log-Lipschitz continuity for the gradient of solutions to the Monge-Amp\`ere…
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.
The regularity theory of the degenerate complex Monge-Amp\`{e}re equation is studied. The equation is considered on a closed compact K\"{a}hler manifold $(M,g)$ with nonnegative orthogonal bisectional curvature of dimension $m$. Given a…
We prove the smoothness of weak solutions to an elliptic complex Monge-Ampere equation, using the smoothing property of the corresponding parabolic flow.
In this work, we study Monge-Ampere equations over closed K\"ahler manifolds with degenerated cohomology classes. Classic results and arguments in pluripotential theory are generalized a little bit to be applied to our situation.
We introduce a parabolic analogue of the elliptic split-type Monge-Amp\`ere equation developed by Fang and the author, extending Streets' twisted Monge-Amp\`ere equation. The resulting equation is fully nonlinear and non-concave. We prove…
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…
We prove the bounded subsolution theorem for the complex Monge-Amp\`ere type equation, with the right hand side being a positive Radon measure, on a compact Hermitian manifold with boundary.
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…
In this paper we investigate the regularity and solvability of solutions to Dirichlet problem for fully non-linear elliptic equations with gradient terms on Hermitian manifolds, which include among others the Monge-Amp\`ere equation for…
A quaternionic version of the Calabi problem was recently formulated by M. Verbitsky and the author. It conjectures a solvability of a quaternionic Monge-Ampere equation on a compact HKT manifold (HKT stays for HyperKaehler with Torsion).…
We establish the uniqueness of solutions to complex Monge-Amp\`ere mean field equations when the temperature parameter is small. In the local setting of bounded hyperconvex domains, our result partially confirms a conjecture by Berman and…
We consider three fundamental classes of compact almost homogeneous manifolds and show that the complements of singular complex orbits in such manifolds are endowed with plurisubharmonic exhaustions satisfying complex homogeneous…
By studying a complex Monge-Amp\`ere equation, we present an alternate proof to a recent result of Chu-Lee-Tam concerning the projectivity of a compact K\"ahler manifold $N^n$ with $\Ric_k< 0$ for some integer $k$ with $1<k<n$, and the…
In this paper, we consider a special relative K\"ahler fibration that satisfies a homogenous Monge-Amp\`ere equation, which is called a Monge-Amp\`ere fibration. There exist two canonical types of generalized Weil-Petersson metrics on the…
For any Hermitian holomorphic vector bundle with Nakano positive curvature tensor, Demailly introduced a Monge-Amp\`ere type equation. When the rank of the bundle is $1$, it becomes the usual Monge-Amp\`ere equation. In this paper, we solve…
The general solution to the Complex Monge-Amp\`ere equation in a two dimensional space is constructed.
We derive a priori estimates for solutions of a general class of fully non-linear equations on compact Hermitian manifolds. Our method is based on ideas that have been used for different specific equations, such as the complex…