Regularity of fully non-linear elliptic equations on Hermitian 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 Kähler 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.

This is the first part of a series of researches devoted to the study of Dirichlet problem for fully non-linear elliptic equations on complex manifolds, which include [arXiv:2001.09238], [arXiv:2106.14837], [Pure Appl. Math. Q. 16 (2020), 1585-1617; MR4221006] and [Calc. Var. PDE. 60 (2021), Paper No. 162, 20 pp.; MR4290375]