Prescribed Scalar Curvature on Compact Manifolds Under Conformal Deformation

TL;DR

This paper establishes conditions under which scalar curvature functions can be prescribed on compact manifolds via conformal deformation, covering cases with boundary, quotients of spheres, and general manifolds, with new results on perturbations and specific conditions.

Contribution

It provides new sufficient and almost necessary conditions for prescribing scalar curvature in conformal classes, including perturbation results and the introduction of CONDITION B for spheres and their quotients.

Findings

Positive functions with certain local properties can be realized as scalar curvatures.

Any smooth function can be approximated by prescribed scalar curvatures via perturbation.

Conditions are identified under which scalar curvature prescription is possible on various manifolds.

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 \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 $ \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.