The Dirichlet problem for Monge-Ampère equation for $(n-1)$-PSH functions on Hermitian manifolds

Abstract

We solve the Dirichlet problem for Monge-Ampère equation for $(n-1)$-PSH functions possibly with degenerate right-hand side, through deriving a quantitative version of boundary estimate under the assumption of $(n-1)$-PSH subsolutions. In addition, we confirm the subsolution assumption on a product of a closed balanced manifold with a compact Riemann surface with boundary.

The main context of this paper is the integration of the second parts of [arXiv:2001.09238] and [arXiv:2106.14837], while [arXiv: 2203.03439], and the first parts of [arXiv:2001.09238] and [arXiv:2106.14837] are reorganized into a separate paper [arXiv:2203.04898]. More precisely, this paper is essentially extracted from Section 7 in [arXiv:2001.09238] and Section 4 in [arXiv:2106.14837]