Prescribed Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree

TL;DR

This paper studies the problem of prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree, providing existence and nonexistence results through geometric flow analysis.

Contribution

It offers new existence and nonexistence results for prescribed Chern scalar curvatures on Hermitian manifolds with negative Gauduchon degree, including sign-changing cases.

Findings

Existence results for nonzero, nonpositive curvature functions.

Nonexistence results for certain sign-changing curvature functions.

Analysis of geometric flow convergence.

Abstract

In this paper, we investigate the problem of prescribing Chern scalar curvatures on compact Hermitian manifolds with negative Gauduchon degree. By studying the convergence of the associated geometric flow, we obtain some existence results when the candidate curvature function is nonzero and nonpositive. Furthermore, we also consider the case that the candidate curvature function is sign-changing, and establish some existence and nonexistence results.