Prescribed Chern scalar curvature flow on compact Hermitian manifolds with negative Gauduchon degree

TL;DR

This paper introduces a flow method to solve the prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree, providing conditions for convergence to a desired curvature.

Contribution

It develops a unified flow approach and establishes sufficient conditions for convergence when the conformal class contains a balanced metric.

Findings

Flow converges to a conformal Hermitian metric with prescribed curvature under certain conditions.

Provides a new method for solving prescribed Chern scalar curvature problems.

Applicable to manifolds with negative Gauduchon degree.

Abstract

In this paper, we present a unified flow approach to prescribed Chern scalar curvature problem on compact Hermitian manifolds with negative Gauduchon degree. When the conformal class of its Hermitian metric contains a balanced metric, we give some sufficient conditions on the candidate curvature function $f$ which guaranties the convergence of the flow to a conformal Hermitian metric whose Chern scalar curvature is $f$.