Prescribed Scalar Curvature on Compact Manifolds Under Conformal Deformation

Abstract

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 ${\mathbb{S}}^n$ or some quotient of ${\mathbb{S}}^n$, in dimensions $n⩾3$, provided that the first eigenvalues of conformal Laplacian (with appropriate boundary conditions if necessary) are positive. When the manifold is not some quotient of ${\mathbb{S}}^n$, 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 ${\mathbb{S}}^n$ or ${\mathbb{S}}^n/\Gamma$ with Kleinian group $\Gamma$ 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.

arXiv comment38 Pages, Oct.10 version added analysis on n-sphere, etc. Dec. 9 version did very few modifications in English