The prescribed Chern scalar curvature problem

TL;DR

This paper investigates the prescribed Chern scalar curvature problem on Hermitian manifolds, providing existence and uniqueness results depending on the Gauduchon degree and the presence of balanced metrics.

Contribution

It offers new solutions and classifications for the prescribed Chern scalar curvature problem based on the sign of the Gauduchon degree and the existence of balanced metrics.

Findings

Unique solutions for negative Gauduchon degree cases.

Existence of solutions for functions with negative mean value on balanced metrics.

Classification of solutions based on the sign of the Gauduchon degree.

Abstract

The paper is an attempt to resolve the prescribed Chern scalar curvature problem. We look for solutions within the conformal class of a fixed Hermitian metric. We divide the problem in three cases, according to the sign of the Gauduchon degree, that we analyse separately. In the case where the Gauduchon degree is negative, we prove that every non-identically zero and non-positive function is the Chern scalar curvature of a unique metric conformal to the fixed one. Moreover, if there exists a balanced metric with zero Chern scalar curvature, we prove that every smooth function changing sign with negative mean value is the Chern scalar curvature of a metric conformal to the balanced one.