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Register a new userThe Complex Sobolev Space and Holder continuous solutions to Monge-Amp`ere equations
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Let $X$ be a compact Kahler manifold of dimension $n$ and $omega$ a Kahler form on $X$. We consider the complex Monge-Amp`ere equation $(dd^c u+omega)^n=mu$, where $mu$ is a given positive measure on $X$ of suitable mass and $u$ is an $omega$-plurisubharmonic function. We show that the equation admits a Holder continuous solution {it if and only if} the measure $mu$, seen as a functional on a complex Sobolev space $W^*(X)$, is Holder continuous. A similar result is also obtained for the complex Monge-Amp`ere equations on domains of $mathbb{C}^n$.
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We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Amp`ere equation on a compact Hermitian manifold for a very general measre on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Holder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.
A new proof for stability estimates for the complex Monge-Amp`ere and Hessian equations is given, which does not require pluripotential theory. A major advantage is that the resulting stability estimates are then uniform under general degenerations of the background metric in the case of the Monge-Amp`ere equation, and under degenerations to a big class in the case of Hessian equations.
We develop a new approach to $L^{infty}$-a priori estimates for degenerate complex Monge-Amp`ere equations on complex manifolds. It only relies on compactness and envelopes properties of quasi-plurisubharmonic functions. In a prequel cite{GL21a} we have shown how this method allows one to obtain new and efficient proofs of several fundamental results in Kahler geometry. In cite{GL21b} we have studied the behavior of Monge-Amp`ere volumes on hermitian manifolds. We extend here the techniques of cite{GL21a} to the hermitian setting and use the bounds established in cite{GL21b}, producing new relative a priori estimates, as well as several existence results for degenerate complex Monge-Amp`ere equations on compact hermitian manifolds.
Let $Omega Subset mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1leq mleq n$) and $mu$ a positive Borel measure on $Omega$. We study the complex Hessian equation $(dd^c u)^m wedge beta^{n - m} = mu$ on $Omega$. First we give a sufficient condition on the measure $mu$ in terms of its domination by the $m$-Hessian capacity which guarantees the existence of a continuous solution to the associated Dirichlet problem with a continuous boundary datum. As an application, we prove that if the equation has a continuous $m$-subharmonic subsolution whose modulus of continuity satisfies a Dini type condition, then the equation has a continuous solution with an arbitrary continuous boundary datum. Moreover when the measure has a finite mass, we give a precise quantitative estimate on the modulus of continuity of the solution. One of the main steps in the proofs is to establish a new capacity estimate showing that the $m$-Hessian measure of a continuous $m$-subharmonic function on $Omega$ with zero boundary values is dominated by an explicit function of the $m$-Hessian capacity with respect to $Omega$, involving the modulus of continuity of $varphi$. Another important ingredient is a new weak stability estimate on the Hessian measure of a continuous $m$-subharmonic function.
We study complex geodesics and complex Monge-Amp`{e}re equations on bounded strongly linearly convex domains in $mathbb C^n$. More specifically, we prove the uniqueness of complex geodesics with prescribed boundary value and direction in such a domain, when its boundary is of minimal regularity. The existence of such complex geodesics was proved by the first author in the early 1990s, but the uniqueness was left open. Based on the existence and the uniqueness proved here, as well as other previously obtained results, we solve a homogeneous complex Monge-Amp`{e}re equation with prescribed boundary singularity, which was first considered by Bracci et al. on smoothly bounded strongly convex domains in $mathbb C^n$.
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