Localized Deformation of the Scalar Curvature and the Mean Curvature

On a compact manifold with boundary, the map consisting of the scalar curvature in the interior and the mean curvature on the boundary is a local surjection at generic metrics. We prove that this result may be localized to compact subdomains in an arbitrary Riemannian manifold with boundary. This result is a generalization of Corvino’s result about localized scalar curvature deformations; however, the existence part needs to be handled delicately since the linearized problem is non-variational. We also discuss generic conditions that guarantee localized deformations, and related geometric properties.