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Extension of symmetries on Einstein manifolds with boundary

Michael T. Anderson

TL;DR

This work establishes conditions under which boundary Killing fields extend to bulk Einstein fillings, linking isometry extension to the divergence constraint and the global topology via $\pi_1(M,\partial M)=0$. The authors develop a gauge-fixed elliptic framework for the Einstein equations, show how surjectivity of the linearized divergence constraint ensures extension, and provide a constructive argument (with a finite-dimensional obstruction handling) to prove Theorem 1.1 and its corollaries. They also address exterior and conformally compact settings, and supply a necessary correction to the initial proof in an Appendix. The results contribute to rigidity phenomena for Einstein manifolds with boundary and clarify when boundary symmetries are inherited by the bulk geometry.

Abstract

We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on the boundary. Under a mild condition on the fundamental group, this is proved to be the case at least when the Killing field preserves the mean curvature of the boundary.

Extension of symmetries on Einstein manifolds with boundary

TL;DR

This work establishes conditions under which boundary Killing fields extend to bulk Einstein fillings, linking isometry extension to the divergence constraint and the global topology via . The authors develop a gauge-fixed elliptic framework for the Einstein equations, show how surjectivity of the linearized divergence constraint ensures extension, and provide a constructive argument (with a finite-dimensional obstruction handling) to prove Theorem 1.1 and its corollaries. They also address exterior and conformally compact settings, and supply a necessary correction to the initial proof in an Appendix. The results contribute to rigidity phenomena for Einstein manifolds with boundary and clarify when boundary symmetries are inherited by the bulk geometry.

Abstract

We investigate the validity of the isometry extension property for (Riemannian) Einstein metrics on manifolds with boundary. Given a metric on the boundary, this is the issue of whether any Killing field of the boundary metric extends to a Killing field of any bulk or filling Einstein metric inducing the given data on the boundary. Under a mild condition on the fundamental group, this is proved to be the case at least when the Killing field preserves the mean curvature of the boundary.

Paper Structure

This paper contains 6 sections, 15 theorems, 220 equations.

Table of Contents

  1. Introduction.
  2. The Space of Einstein Metrics
  3. Elliptic Boundary Value Problems for the Einstein Equations
  4. Isometry Extension and the Divergence Constraint
  5. Proof of the Main Results.
  6. Appendix: Corrections (October, 2012)

Key Result

Theorem 1.1

Let $g$ be a $C^{m,\alpha}$ Einstein metric on $M$, $m \geq 5$, with induced boundary metric $\gamma$ on $\partial M$, and suppose e1.2 holds. Then any Killing field $X$ on $(\partial M, \gamma)$ for which $X(H) = 0$, extends uniquely to a Killing field on $(M, g)$.

Theorems & Definitions (24)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Remark 3.1
  • Proposition 3.2
  • Remark 3.3
  • ...and 14 more