Regularity of fully non-linear elliptic equations on Hermitian manifolds

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.