The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

TL;DR

This paper investigates the prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds, reducing it to a semi-linear PDE and establishing existence conditions based on the sign of a related constant, also connecting to other Yamabe problems.

Contribution

It introduces a new approach to solving the prescribed Gauduchon scalar curvature problem via conformal variations and PDE analysis, extending to related Yamabe problems in Hermitian geometry.

Findings

Existence of solutions depends on the sign of a Gauduchon degree-related constant.

Provides necessary and sufficient conditions for prescribed functions to be Gauduchon scalar curvatures.

Recovers classical Chern and Bismut Yamabe problems within this framework.

Abstract

In this paper we consider the prescribed Gauduchon scalar curvature problem on almost Hermitian manifolds. By deducing the expression of the Gauduchon scalar curvature under the conformal variation, the problem is reduced to solve a semi-linear partial differential equation with exponential nonlinearity. Using super and sub-solution method, we show that the existence of the solution to this semi-linear equation depends on the sign of a constant associated to Gauduchon degree. When the sign is negative, we give both necessary and sufficient conditions that a prescribed function is the Gauduchon scalar curvature of a conformal Hermitian metric. Besides, this paper recovers Chern Yamabe problem, prescribed Chern Yamabe problem and Bismut Yamabe problem.