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Deformations of the Scalar Curvature and the Mean Curvature

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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, as motivated by an attempt to generalize the Riemannian Penrose inequality in dimension 8. 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 problem is non-variational. For non-generic cases, we give a classification theorem for domains in space forms and Schwarzschild manifolds, and show the connection with positive mass theorems.

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This item is under embargo until June 27, 2024.