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Localized Deformation of the Scalar Curvature and the Mean Curvature

Hongyi Sheng

TL;DR

This work develops a boundary-localized gluing theory for the Einstein constraint system by studying the coupled map $(R,H)$ that prescribes interior scalar curvature and boundary mean curvature. Using a Fredholm framework in weighted Sobolev and Hölder spaces, the authors decompose tensor deformations into a principal component and a finite-dimensional complement, and prove Poincaré-type and weighted Schauder estimates to control regularity. Under generic conditions (notably the triviality of the adjoint kernel), they establish local surjectivity of the linearized map and extend it to a nonlinear deformation via a Picard iteration, achieving localized realizations of prescribed curvature near boundary patches. The results generalize Corvino's interior scalar-curvature localization to manifolds with boundary, with implications for boundary gluing, density theorems for asymptotically flat scalar-flat manifolds, and regularity issues in area-minimizing hypersurfaces within initial data sets.

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.

Localized Deformation of the Scalar Curvature and the Mean Curvature

TL;DR

This work develops a boundary-localized gluing theory for the Einstein constraint system by studying the coupled map that prescribes interior scalar curvature and boundary mean curvature. Using a Fredholm framework in weighted Sobolev and Hölder spaces, the authors decompose tensor deformations into a principal component and a finite-dimensional complement, and prove Poincaré-type and weighted Schauder estimates to control regularity. Under generic conditions (notably the triviality of the adjoint kernel), they establish local surjectivity of the linearized map and extend it to a nonlinear deformation via a Picard iteration, achieving localized realizations of prescribed curvature near boundary patches. The results generalize Corvino's interior scalar-curvature localization to manifolds with boundary, with implications for boundary gluing, density theorems for asymptotically flat scalar-flat manifolds, and regularity issues in area-minimizing hypersurfaces within initial data sets.

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.
Paper Structure (23 sections, 55 theorems, 258 equations, 3 figures)

This paper contains 23 sections, 55 theorems, 258 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notation
  4. Linearized equations and integration by parts formula
  5. Approximate distance and weight function
  6. Weighted Sobolev spaces
  7. Weighted Hölder spaces
  8. A basic lemma
  9. Generic conditions
  10. The kernels of $\Psi^*$ and $\Phi^*$
  11. Generic conditions
  12. Localized Deformation
  13. Poincaré-type inequalities
  14. The Dirichlet problems and the space $\mathcal{D}$
  15. The self-adjoint problem
  16. ...and 8 more sections

Key Result

Theorem 1.1

Let $M^n (n \ge 2)$ be a Riemannian manifold (not necessarily compact) with boundary $\partial M$ and a $\mathcal{C}^{k+4,\alpha}$-metric $g_0$. Let $\overline{\Omega^n}\subset\overline{M}$ be a compact, connected subdomain with $C^{k+4,\alpha}$ boundary such that: Assume the generic conditions from Section sec3 hold. Then there exists $\epsilon > 0$ such that for all pairs $(R',H')$ satisfying:

Figures (3)

  • Figure 1: Smale's perturbation and localized deformation
  • Figure 2: Collar neighborhood of $\Sigma'$
  • Figure 3: Level sets of $\theta$ and $\bar{d}$ in Fermi coordinates

Theorems & Definitions (92)

  • Theorem 1.1
  • Proposition 2.1
  • Proposition 2.2: Green’s formula
  • Lemma 2.3: Corvino-Schoen C-S*Lemma 2.1
  • Proposition 2.4
  • proof
  • Corollary 2.5
  • Corollary 2.6
  • Lemma 2.7
  • proof
  • ...and 82 more