Fully non-linear elliptic equations on noncompact complex manifolds

TL;DR

This paper develops a framework for solving fully non-linear elliptic equations on noncompact complex manifolds, leading to new geometric structures such as complete Kähler metrics and Einstein metrics.

Contribution

It provides a priori estimates and existence results for these equations, enabling the construction of specific geometric metrics on noncompact manifolds.

Findings

Constructed complete Kähler metrics with prescribed volume forms

Established existence of Einstein metrics on noncompact Kähler manifolds

Developed a priori estimates for fully non-linear equations

Abstract

In this paper, we establish a priori estimates and existence results for solutions of a general class of fully non-linear equations on noncompact K\"{a}hler and Hermitian manifolds. As geometric applications, we construct complete K\"{a}hler metrics with prescribed volume forms on strictly pseudoconvex domains, as well as find Einstein metrics on complete noncompact K\"{a}hler manifolds and Hessian manifolds with negative first Chern class.