A Monge-Ampère type fully nonlinear equation on Hermitian manifolds

Abstract

We study a fully nonlinear equation of complex Monge-Ampère type on Hermitian manifolds. We establish the a priori estimates for solutions of the equation up to the second order derivatives with the help of a subsolution.1991 Mathematics Subject Classification. 58J05, 58J32, 32W20, 35J25, 53C55.

Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds

“…Similar inequalities for f = σ n /σ n−1 were previously used by Song and Weinkove [38] in their work on the J-flow on Käher manifolds, and by Fang et al [9] who extended the result of [38] to f = (σ n /σ k ) 1 n−k for 1 ≤ k ≤ n − 1; see also [16][17][18]. Note that by the concavity of f we always have…”

A criterion for the properness of the $$K$$ K -energy in a general Kähler class

“…In Remark 1.3 of [9], they asked whether the critical points of those fully nonlinear flows can be solved by using the elliptic method instead of the geometric flow method. Later, in a series of papers [13] [14][15] Guan and his collaborators gave some C 2 estimates for these critical points on Hermitian manifolds. Then Sun proved Song-Weinkove's result by the elliptic method on general Hermitian manifolds [26].…”

On a class of fully nonlinear elliptic equations on Hermitian manifolds

“…In this paper we prove the following existence result which is new even in the case when M is a bounded domain in C n and χ = 0; we assume 2 ≤ α ≤ n − 2 as the cases α = 1 and α = n − 1 were considered in [17] and [18], while for the complex Monge-Ampère equation (α = n) it was proved in [16].…”