Fully nonlinear elliptic equations for some prescribed curvature problems on Hermitian manifolds

TL;DR

This paper investigates fully nonlinear elliptic equations on Hermitian manifolds, establishing conditions for conformal metrics with prescribed Chern-Ricci curvature, and provides geometric insights with near-optimal assumptions.

Contribution

It introduces new methods for solving nonlinear elliptic equations on Hermitian manifolds and derives geometric consequences for prescribed curvature problems.

Findings

Established partial uniform ellipticity for the equations.

Derived conditions for existence of conformal Hermitian metrics.

Identified near-sharp assumptions due to geometric obstructions.

Abstract

We study fully nonlinear elliptic equations on Hermitian manifolds through blow-up argument and partial uniform ellipticity. We apply our results to draw geometric conclusions on finding conformal Hermitian metrics with prescribed Chern-Ricci curvature functions. By some obstruction from geometric function theory, our assumptions are almost sharp.