Abstract:
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 irreducible singular K\"ahler spaces with ample or trivial canonical sheaf and singular K\"ahler-Einstein metrics over varieties of general type.

Abstract:
We study various capacities on compact K\"{a}hler manifolds which generalize the Bedford-Taylor Monge-Amp\`ere capacity. We then use these capacities to study the existence and the regularity of solutions of complex Monge-Amp\`ere equations.

Abstract:
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 $\inf_M \Delta f^{1/(m-1)}$.

Abstract:
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 compact setting results of U.Cegrell from the local pluripoltential theory. We give some applications to complex dynamics and to the existence of K\"ahler-Einstein metrics on singular manifolds.

Abstract:
We first obtain the interior $C^{1,1}$-regularity and solvability for the degenerate real Monge-Amp\`ere equation in a bounded, $C^3$-smooth and strictly convex domain in $\mathbb R^d$ ($d\ge2$), assuming that the boundary data is only globally $C^{1,1}$, and the $d$-th root of the nonnegative right-hand side is globally $C^{0,1}$ and convex after adding $K|x|^2$ for some constant $K$. Then we establish the interior $C^{1,1}$-regularity and solvability for the degenerate complex Monge-Amp\`ere equation in a bounded, $C^3$-smooth and strictly pseudoconvex domain in $\mathbb C^d$, under the global $C^{1,1}$-regularity assumption on the boundary data and the $d$-th root of the nonnegative right-hand side. Since the derivatives may blow up along non-tangent directions at the boundary under our regularity assumptions on the boundary data, we also estimate the derivatives up to second order in both problems. Our technique is probabilistic by following Krylov's approach. The result in the real case extends N. Trudinger and J. Urbas's interior $C^{1,1}$-regularity result for the homogeneous case [N. Trudinger and J. Urbas, Bull. Austral. Math. Soc., 30(3): 321~334, 1984.] in the sense of considering the nonnegative right-hand side. The result in the complex case generalizes E. Bedford and B. A. Taylor's interior $C^{1,1}$-regularity result in a ball [E. Bedford and B. A. Taylor, Invent. Math., 37(1): 1~44, 1976.] by allowing the domain be any bounded, sufficiently smooth and strictly pseudoconvex one.

Abstract:
We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^2u \leq 1, \quad u|_{\partial \Omega} = 0$$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.

Abstract:
Let $(X,\omega)$ be a compact K\"ahler manifold of dimension $n$ and fix $m\in \mathbb{N}$ such that $1\leq m \leq n$. We prove that any $(\omega,m)$-sh function can be approximated from above by smooth $(\omega,m)$-sh functions. A potential theory for the complex Hessian equation is also developed which generalizes the classical pluripotential theory on compact K\"ahler manifolds. We then use novel variational tools due to Berman, Boucksom, Guedj and Zeriahi to study degenerate complex Hessian equations.

Abstract:
A C^2 function on C^n is called (n-1)-plurisubharmonic in the sense of Harvey-Lawson if the sum of any n-1 eigenvalues of its complex Hessian is nonnegative. We show that the associated Monge-Ampere equation can be solved on any compact Kahler manifold. As a consequence we prove the existence of solutions to an equation of Fu-Wang-Wu, giving Calabi-Yau theorems for balanced, Gauduchon and strongly Gauduchon metrics on compact Kahler manifolds.

Abstract:
We consider degenerate Monge-Ampere equations of the type $$\det D^2 u= f \quad \{in $\Om$}, \quad \quad f \sim \, d_{\p \Om}^\alpha \quad \{near $\p \Om$,}$$ where $d_{\p \Om}$ represents the distance to the boundary of the domain $\Om$ and $\alpha>0$ is a positive power. We obtain $C^2$ estimates at the boundary under natural conditions on the boundary data and the right hand side. Similar estimates in two dimensions were obtained by J.X. Hong, G. Huang and W. Wang.

Abstract:
We prove that on compact K\"ahler manifolds solutions to the complex Monge-Amp\`ere equation, with the the right hand side in $L^p, p>1,$ are H\"older continuous.