2022/05/30 by Xu, Jie
#35J60 #53C18 #58J05 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics
paper · doi:10.48550/arxiv.2205.15453
We give sufficient and "almost" necessary conditions for the prescribed scalar curvature problems within the conformal class of a Riemannian metric g for both closed manifolds and compact manifolds with boundary, including the interesting cases \mathbbSn or some quotient of \mathbbSn , in dimensions n \geqslant 3 , provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of \mathbbSn , we show that, on one hand, any smooth function that is a positive constant within some open subset of the manifold with arbitrary positive measure, and has no restriction on the rest of the manifold, is a prescribed scalar curvature function of some metric under conformal change; on the other hand, any smooth function S is almost a prescribed scalar curvature function of Yamabe metric within the conformal class [g] in the sense that an appropriate perturbation of S that defers with S within an arbitrarily small open subset is a prescribed scalar curvature function of Yamabe metric. When the manifold is either \mathbbSn or \mathbbSn / Γ with Kleinian group Γ we show that any positive function that satisfies a technical analytical condition, called CONDITION B, can be realized as a prescribed scalar curvature functions on these manifolds.