Regularity of fully non-linear elliptic equations on Hermitian manifolds

TL;DR

This paper develops new boundary estimates for fully non-linear elliptic equations on Hermitian manifolds, enabling unified analysis of existence and regularity of solutions, including applications to Kähler geometry.

Contribution

It introduces a novel approach to boundary estimates that applies to complex and real manifolds, advancing the understanding of fully non-linear elliptic equations.

Findings

Established quantitative boundary estimates for solutions.

Unified approach to existence and regularity of solutions.

Applicable to both Hermitian and Riemannian manifolds.

Abstract

In this paper we propose new insights and ideas to set up quantitative boundary estimates for solutions to Dirichlet problem of a class of fully non-linear elliptic equations on compact Hermitian manifolds with real analytic Levi flat boundary. With the quantitative boundary estimates at hand, we can establish the gradient estimate and give a unified approach to investigate the existence and regularity of solutions of Dirichlet problem with sufficiently smooth boundary data, which include the geodesic equation in the space of K\"ahler metrics as a special case. Our method can also be applied to Dirichlet problem for analogous fully non-linear elliptic equations on a compact Riemannian manifold with concave boundary.