The Dirichlet problem for Monge-Amp\`ere equation for $(n-1)$-PSH functions on Hermitian manifolds

TL;DR

This paper addresses solving the Dirichlet problem for the Monge-Ampère equation involving (n-1)-plurisubharmonic functions on Hermitian manifolds, providing boundary estimates and confirming subsolution assumptions in specific geometric contexts.

Contribution

It introduces a boundary estimate for the Dirichlet problem with degenerate data and verifies subsolution conditions on certain product manifolds, advancing understanding of complex Monge-Ampère equations.

Findings

Established a quantitative boundary estimate for the problem.

Confirmed subsolution existence on product manifolds with boundary.

Extended solvability to degenerate right-hand sides.

Abstract

We solve the Dirichlet problem for Monge-Amp\`ere 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.