The twisted constant in Calabi-Yau type equation

TL;DR

This paper provides a comprehensive condition for solving nonlinear elliptic equations on almost Hermitian manifolds and determines the twisted constants in various Calabi-Yau type equations, addressing open questions in the field.

Contribution

It extends previous work by establishing a necessary and sufficient condition for these equations and explicitly determines the twisted constants in several important cases.

Findings

Established a criterion for solving fully nonlinear elliptic equations on almost Hermitian manifolds.

Determined the twisted constants in classical, Hessian, and form type Calabi-Yau equations.

Addressed an open question from prior research by Chu-Tosatti-Weinkove.

Abstract

In this paper we establish a necessary and sufficient condition for solving a general class of fully nonlinear elliptic equations on compact almost Hermitian manifolds, extending a recent work of Guo-Song. As applications, we determine the twisted constants in Calabi-Yau type equations, including the classical one, Hessian type one and form type one with gradient terms introduced by Popovici and Tosatti-Weinkove. In particular, it addresses a question raised in a work of Chu-Tosatti-Weinkove \cite[Introduction, Remark 5]{CTW19}.