Subsolution theorem for the MongeAmpère equation over almost Hermitian manifold
Abstract
Let $\Omega\subseteq M$ be a bounded domain with smooth boundary $\partial\Omega$, where $(M,J,g)$ is a compact almost Hermitian manifold. Our main result of this paper is to consider the Dirichlet problem for complex MongeAmpère 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 Pliś when we assume there is a strictly $J$psh defining function for $\Omega$.
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2107.00167
 Bibcode:
 2021arXiv210700167Z
 Keywords:

 Mathematics  Analysis of PDEs;
 Mathematics  Differential Geometry;
 32W20;
 32Q60;
 35B50;
 31C10
 EPrint:
 28 pages. All comments are welcome!