Prescribed Scalar Curvature on Compact Manifolds Under Conformal Deformation

Jie Xu

Published 2022-05-30Version 1

We give results of sufficient and ``almost" necessary conditions of prescribed scalar curvature problems under 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. 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 of some metric under conformal change; on the other hand, any smooth function $ S $ is almost a prescribed scalar curvature of some metric within the conformal class in the sense that an appropriate perturbation of $ S $ within arbitrarily small open subset is a prescribed scalar curvature of some metric by pointwise conformal deformation. The original function $ S $ and its perturbation only disagree within an arbitrarily small set.