Regularity of fully non-linear elliptic equations on Hermitian manifolds. II

TL;DR

This paper studies the regularity and solvability of fully non-linear elliptic equations, including Monge-Ampère, on Hermitian manifolds, revealing new boundary regularity features influenced by boundary shape.

Contribution

It introduces novel boundary regularity assumptions and estimates for non-linear elliptic equations on Hermitian manifolds, extending previous results and applying blow-up techniques.

Findings

New boundary regularity assumptions derived from boundary shape.

Quantitative boundary estimates enabling gradient bounds.

Construction of subsolutions on product manifolds.

Abstract

In this paper we investigate the regularity and solvability of solutions to Dirichlet problem for fully non-linear elliptic equations with gradient terms on Hermitian manifolds, which include among others the Monge-Amp\`ere equation for $(n-1)$-plurisubharmonic functions. Some significantly new features of regularity assumptions on the boundary and boundary data are obtained, which reveal how the shape of the boundary influences such regularity assumptions. Such new features follow from quantitative boundary estimates which specifically enable us to apply a blow-up argument to derive the gradient estimate. Interestingly, the subsolutions are constructed when the background space is moreover a product of a closed Hermitian manifold with a compact Riemann surface with boundary.