Continuity of the Complex Monge-Ampere Operator on Compact Kahler Manifolds

Let $(X,\omega)$ be a compact Hermitian manifold of complex dimension $n$. Let $\beta$ be a smooth real closed $(1,1)$ form such that there exists a function $\rho \in \mbox{PSH}(X,\beta)\cap L^{\infty}(X)$. We study the range of the complex non-pluripolar Monge-Amp\`ere operator $\langle(\beta+dd^c\cdot)^n\rangle$ on weighted Monge-Amp\`ere energy classes on $X$. In particular, when $\rho$ is assumed to be continuous, we give a complete characterization of the range of the c...

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.

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 ampleness of the canonical line bundle $K_N$.

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.

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 solution $\phi$ of the degenerate complex Monge-Amp\`{e}re equation $\det(g_{i \bar{j}} + \phi_{i \bar{j}}) = f \det(g_{i \bar{j}})$, it is shown that the Laplacian of $\phi$ can be controlled by a constant depending on $(M,g)$, $\sup f$, and ...

Let $(X,\omega)$ be a compact K\"ahler manifold. We introduce and study the largest set $DMA(X,\omega)$ of $\omega$-plurisubharmonic (psh) functions on which the complex Monge-Amp\`ere operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\om)$ of all $\om$-psh functions. We prove that certain twisted Monge-Amp\`ere operators are well defined for all $\omega$-psh functions. As a consequenc...

In this paper, we obtain the Bedford-Taylor interior $C^{2}$ estimate and local Calabi $C^{3}$ estimate for the solutions to complex Monge-Amp\`ere equations on Hermitian manifolds.

We establish various stability results for solutions of complex Monge-Amp\`ere equations in big cohomology classes, generalizing results that were known to hold in the context of K\"ahler classes.

We prove uniform gradient and diameter estimates for a family of geometric complex Monge-Ampere equations. Such estimates can be applied to study geometric regularity of singular solutions of complex Monge-Ampere equations. We also prove a uniform diameter estimate for collapsing families of twisted Kahler-Einstein metrics on Kahler manifolds of nonnegative Kodaira dimensions.

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.