Sup-slopes and sub-solutions for fully nonlinear elliptic equations

TL;DR

This paper provides a comprehensive criterion for solving a broad class of fully nonlinear elliptic equations on complex manifolds, addressing an open problem and applying the results to complex geometric equations.

Contribution

It establishes a necessary and sufficient condition for solving these equations, based on an analytic slope invariant, advancing the understanding of complex geometric PDEs.

Findings

Solved the open problem of Li and Urbas.

Established a criterion based on slope invariants.

Applied results to the non-constant J-equation on complex manifolds.

Abstract

We establish a necessary and sufficient condition for solving a general class of fully nonlinear elliptic equations on closed Riemannian or hermitian manifolds, including both hessian and hessian quotient equations. It settles an open problem of Li and Urbas. Such a condition is based on an analytic slope invariant analogous to the slope stability and the Nakai-Moishezon criterion in complex geometry. As an application, we solve the non-constant $J$-equation on both hermitian manifolds and singular K\"ahler spaces.