Prescribed Scalar Curvature on Compact Manifolds Under Conformal Deformation

We give results of sufficient and "almost" necessary conditions of prescribed scalar curvature problems under pointwise conformal change on closed manifolds and compact manifolds with boundary 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 $ \mathbb{S}^{n} $ or some quotient of $ \mathbb{S}^{n} $, we show that, on one hand, any smooth function that is equal to some 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 some metric within the conformal class 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 $ \mathbb{S}^{n} $ or the quotient of $ \mathbb{S}^{n} $ by some group $ \Gamma $ that relates to the Kleinain group, we show that any positive function that satisfies a special analytical condition, the CONDITION B, can be realized as a prescribed scalar curvature functions on these manifolds.Comment: 38 Pages, Oct. 10 version added analysis on n-sphere, et