Related papers: Degenerate Complex Monge Ampere Equation Part I
A PDE proof is provided for the sharp $L^\infty$ estimates for the complex Monge-Amp\`ere equation which had required pluripotential theory before. The proof covers both cases of fixed background as well as degenerating background metrics.…
In this paper, we consider a class of Hessian type equations which include the $(n-1)$ Monge-Amp\`{e}re equation on Riemannian manifolds. The \emph{a priori} $C^2$ estimates and the existence of solutions are established.
This is a contribution to the special issue of Surveys in Differential Geometry celebrating the 75th birthday of Shing-Tung Yau. The bulk of the paper is devoted to a survey of some new geometric inequalities and estimates for the…
We discuss Monge-Amp\`ere equations from the view point of differential geometry. It is known that a Monge-Amp\`ere equation corresponds to a special exterior differential system on a 1-jet space. In this paper, we generalize Monge-Amp\`ere…
We consider a general class of elliptic equations on hypercomplex manifolds which includes the quaternionic Monge-Amp\`ere equation, the quaternionic Hessian equation and the Monge-Amp\`ere equation for quaternionic $(n-1)$-plurisubharmonic…
A quaternionic version of the Calabi problem on Monge-Ampere equation is introduced. It is a quaternionic Monge-Ampere equation on a compact hypercomplex manifold with an HKT-metric. The equation is non-linear elliptic of second order. For…
By using the global deformation of almost complex structures which are compatible with a symplectic form off a Lebesgue measure zero subset, we construct a (measurable) Lipschitz Kahler metric such that the one-form type Calabi-Yau equation…
We introduce the concept of k-strictly convexity to describe the accurate convexity of convex domains some directions of which boundary may be flat. Basing this accurate convexity, we construct sub-solutions the Dirichlet problem for some…
We define a general class of elliptic equations for 2-forms on 4-manifolds, of which the complex Monge-Ampere equation is a prototype. We obtain some regularity results and discuss various connections (some speculative) with modern…
By constructing explicit supersolutions, we obtain the optimal global H\"older regularity for several singular Monge-Amp\`ere equations on general bounded open convex domains including those related to complete affine hyperbolic spheres,…
A (bounded) manifold of circular type is a complex manifold M of dimension n admitting a (bounded) exhaustive real function u, defined on M minus a point x_o, so that: a) it is a smooth solution on $M\setminus {x_o}$ to the Monge-Amp\`ere…
For any $\theta<\frac{1}{3}$, we show that very weak solutions to the two-dimensional Monge-Amp\`ere equation with regularity $C^{1,\theta}$ are dense in the space of continuous functions. This result is shown by a convex integration scheme…
Extending a recent theory developed on compact K\"ahler manifolds by Guedj-Lu-Zeriahi and the author, we define and study pluripotential solutions to degenerate parabolic complex Monge-Amp\`ere equations on compact Hermitian manifolds.…
The present paper is concerned a class of quasi-linear elliptic degenerate equations. The degenerate operator comes from the analysis of manifolds with corner singularity. Variational methods are applied to verify the existence of infinity…
We develop the first steps of a parabolic pluripotential theory in bounded strongly pseudo-convex domains of Cn. We study certain degenerate parabolic complex Monge-Amp{\`e}re equations, modelled on the K{\"a}hler-Ricci flow evolving on…
In this paper, we establish the Gevrey regularity of solutions for a class of degenerate Monge-Amp\`ere equations in the plane, under the assumption that one principle entry of the Hessian is strictly positive and an appropriately finite…
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…
Singular degenerate differential operator equations are studied. The uniform separability of boundary value problems for degenerate elliptic equation and optimal regularity properties of Cauchy problem for degenerate parabolic equation are…
We show that on any weakly pseudoconvex $B$-regular domain, the classical Dirichlet problem for the complex Monge--Amp\`ere equation with $\mathcal{C}^\infty$-smooth data does not in general admit $\mathcal{C}^{1,1}$-smooth solutions. 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…