Localized Deformation of the Scalar Curvature and the Mean Curvature

Hongyi Sheng

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arXiv · 2402.08619

Localized Deformation of the Scalar Curvature and the Mean Curvature

Hongyi Sheng

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Abstract

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.

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BibTeX RIS

Hongyi Sheng. 2024-01-12. Localized Deformation of the Scalar Curvature and the Mean Curvature. https://arxiv.org/abs/2402.08619

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