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 proble...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.展开更多
In this paper,we study the existence of conformal metrics with the constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed,connected almost Hermitian manifolds of di...In this paper,we study the existence of conformal metrics with the constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed,connected almost Hermitian manifolds of dimension n>6.In addition,we obtain an application and a variational formula for the associated conformal invariant.展开更多
We give a survey on various results regarding the metric aspects of conic surfaces with emphasis on the prescribing curvature problem for conic surfaces.
基金supported by National Natural Science Foundation of China(Grant No.11701426)supported by National Natural Science Foundation of China(Grant No.11501505)。
文摘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.
基金supported by Beijing Natural Science Foundation(Grant No.Z190003)National Natural Science Foundation of China(Grant Nos.12171037 and 12271040)the Fundamental Research Funds for the Central Universities.
文摘In this paper,we study the existence of conformal metrics with the constant holomorphic d-scalar curvature and the prescribed holomorphic d-scalar curvature problem on closed,connected almost Hermitian manifolds of dimension n>6.In addition,we obtain an application and a variational formula for the associated conformal invariant.
基金Acknowledgements This paper grows out from the first Chinese-German Workshop on Metric Riemannian Geometry held in October 2015 at Shanghai Jiao Tong University. The author wishes to thank organizers for kind invitation and encouragement to write this survey. This work was partially supported by the Shanghai Sailing Program （No. 15YF1406200） and the National Natural Science Foundation of China （Grant No. 11501360）.
文摘We give a survey on various results regarding the metric aspects of conic surfaces with emphasis on the prescribing curvature problem for conic surfaces.
