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Unique continuation results for Ricci curvature and applications

Michael T. Anderson

TL;DR

The paper studies boundary behavior problems for metrics with prescribed Ricci curvature on a compact manifold with boundary, examining two classes: metrics that extend smoothly to the closure with boundary metric gamma = g|∂M and conformally compact metrics for which g̃ = ρ² g extends to ∂M so only the conformal class [gamma] is determined. It analyzes how Cauchy data for Ric_g = 0 near the boundary depends on the choice of coordinates and shows that H-harmonic coordinates preserve the Cauchy data and enable unique continuation via Calderón theory. By constructing an H-harmonic foliation and a closed elliptic system for (g_{ij}, u, σ), the metric is determined from boundary data; in particular, two Einstein metrics with identical boundary Cauchy data must coincide locally and, by analytic continuation, globally. The results extend to conformally compact Einstein metrics: with common boundary metric and a structural condition, the metrics are isometric on the domain, with the Fefferman–Graham expansion and Mazzeo’s unique continuation upgrading boundary agreement to global equality and addressing infinitesimal deformations under a topological constraint. Finally, the paper derives boundary-to-interior rigidity for complete conformally compact Einstein metrics via the Gauss–Codazzi constraints, showing that boundary Killing fields extend to bulk Killing fields under suitable topological hypotheses, and discusses obstructions in notable examples such as AdS Schwarzschild spaces.

Abstract

Final version in paper linked above.

Unique continuation results for Ricci curvature and applications

TL;DR

The paper studies boundary behavior problems for metrics with prescribed Ricci curvature on a compact manifold with boundary, examining two classes: metrics that extend smoothly to the closure with boundary metric gamma = g|∂M and conformally compact metrics for which g̃ = ρ² g extends to ∂M so only the conformal class [gamma] is determined. It analyzes how Cauchy data for Ric_g = 0 near the boundary depends on the choice of coordinates and shows that H-harmonic coordinates preserve the Cauchy data and enable unique continuation via Calderón theory. By constructing an H-harmonic foliation and a closed elliptic system for (g_{ij}, u, σ), the metric is determined from boundary data; in particular, two Einstein metrics with identical boundary Cauchy data must coincide locally and, by analytic continuation, globally. The results extend to conformally compact Einstein metrics: with common boundary metric and a structural condition, the metrics are isometric on the domain, with the Fefferman–Graham expansion and Mazzeo’s unique continuation upgrading boundary agreement to global equality and addressing infinitesimal deformations under a topological constraint. Finally, the paper derives boundary-to-interior rigidity for complete conformally compact Einstein metrics via the Gauss–Codazzi constraints, showing that boundary Killing fields extend to bulk Killing fields under suitable topological hypotheses, and discusses obstructions in notable examples such as AdS Schwarzschild spaces.

Abstract

Final version in paper linked above.

Paper Structure

This paper contains 5 sections, 15 theorems, 188 equations.

Table of Contents

  1. Introduction.
  2. Local Coordinates and Cauchy Data
  3. Proof of Theorem 1.1
  4. Proof of Theorem 1.2.
  5. Isometry Extension and the Constraint Equations.

Key Result

Theorem 1.1

Let (M, g) be a C^{2,\alpha} metric on a compact manifold with boundary M, with induced metric \gamma = g|_{\partial M}, and let A be the 2^{\rm nd} fundamental form of \partial M in M. Suppose the Ricci curvature Ric_{g} satisfies where \lambda is a fixed constant. Then (M, g) is uniquely determined up to local isometry by the Cauchy data (\gamma, A) on an arbitrary open set U of \partial M. In

Theorems & Definitions (23)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Remark 3.5
  • Proposition 3.6
  • ...and 13 more