TY - JOUR

T1 - The prescribed Gauduchon scalar curvature problem in almost Hermitian geometry

AU - Li, Yuxuan

AU - Zhou, Wubin

AU - Zhou, Xianchao

N1 - Publisher Copyright:
© 2023, Science China Press.

PY - 2023

Y1 - 2023

N2 - 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, we reduce the problem to solving a semi-linear partial differential equation with exponential nonlinearity. Using the super- and sub-solutions method, we show that the existence of the solution to this semi-linear equation depends on the sign of a constant associated with the Gauduchon degree. When the sign is negative, we give both necessary and sufficient conditions such that a prescribed function is the Gauduchon scalar curvature of a conformal Hermitian metric. Besides, this paper recovers the Chern-Yamabe problem, the Lichnerowicz-Yamabe problem, and the Bismut-Yamabe problem.

AB - 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, we reduce the problem to solving a semi-linear partial differential equation with exponential nonlinearity. Using the super- and sub-solutions method, we show that the existence of the solution to this semi-linear equation depends on the sign of a constant associated with the Gauduchon degree. When the sign is negative, we give both necessary and sufficient conditions such that a prescribed function is the Gauduchon scalar curvature of a conformal Hermitian metric. Besides, this paper recovers the Chern-Yamabe problem, the Lichnerowicz-Yamabe problem, and the Bismut-Yamabe problem.

KW - 53C18

KW - 53C56

KW - almost Hermitian manifold

KW - Gauduchon connection

KW - prescribed scalar curvature problem

UR - http://www.scopus.com/inward/record.url?scp=85171862091&partnerID=8YFLogxK

U2 - 10.1007/s11425-023-2179-0

DO - 10.1007/s11425-023-2179-0

M3 - Article

AN - SCOPUS:85171862091

SN - 1674-7283

JO - Science China Mathematics

JF - Science China Mathematics

ER -