Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSubsolution theorem for the Monge-Amp`{e}re equation over almost Hermitian manifold
155
0
0.0
(
0
)
Authors
Jiaogen Zhang
Ask ChatGPT about the research
No Arabic abstract
Let $Omegasubseteq M$ be a bounded domain with smooth boundary $partialOmega$, where $(M,J,g)$ is a compact almost Hermitian manifold. Our main result of this paper is to consider the Dirichlet problem for complex Monge-Amp`{e}re equation on $Omega$. Under the existence of a $C^{2}$-smooth strictly $J$-plurisubharmonic ($J$-psh for short) subsolution, we can solve this Dirichlet problem. Our method is based on the properties of subsolution which have been widely used for fully nonlinear elliptic equations over Hermitian manifolds. %This work was already done by Plis when we assume there is a strictly $J$-psh defining function for $Omega$.
rate research
Read More
In this paper we consider the Monge-Amp`{e}re type equations on compact almost Hermitian manifolds. We derive a priori estimates under the existence of an admissible $mathcal{C}$-subsolution. Finally, we also obtain an existence theorem if there exists an admissible supersolution.
We review recent advances in the numerical analysis of the Monge-Amp`ere equation. Various computational techniques are discussed including wide-stencil finite difference schemes, two-scaled methods, finite element methods, and methods based on geometric considerations. Particular focus is the development of appropriate stability and consistency estimates which lead to rates of convergence of the discrete approximations. Finally we present numerical experiments which highlight each method for a variety of test problem with different levels of regularity.
Let $(X, omega)$ be a compact Kahler manifold of complex dimension n and $theta$ be a smooth closed real $(1,1)$-form on $X$ such that its cohomology class ${ theta }in H^{1,1}(X, mathbb{R})$ is pseudoeffective. Let $varphi$ be a $theta$-psh function, and let $f$ be a continuous function on $X$ with bounded distributional laplacian with respect to $omega$ such that $varphi leq f. $ Then the non-pluripolar measure $theta_varphi^n:= (theta + dd^c varphi)^n$ satisfies the equality: $$ {bf{1}}_{{ varphi = f }} theta_varphi^n = {bf{1}}_{{ varphi = f }} theta_f^n,$$ where, for a subset $Tsubseteq X$, ${bf{1}}_T$ is the characteristic function. In particular we prove that [ theta_{P_{theta}(f)}^n= { bf {1}}_{{P_{theta}(f) = f}} theta_f^nqquad {rm and }qquad theta_{P_theta[varphi](f)}^n = { bf {1}}_{{P_theta[varphi](f) = f }} theta_f^n. ]
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.
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.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
