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A relative mass cocycle and the mass of asymptotically hyperbolic manifolds

Andreas Cap, A. Rod Gover

Abstract

We construct a cocycle that, for a given $n$-manifold, maps pairs of asymptotically locally hyperbolic (ALH) metrics to a tractor-valued $(n-1)$-form field on the conformal infinity. This requires the metrics to be asymptotically related to a given order that depends on the dimension. It then provides a local geometric quantity on the boundary that is naturally associated to the pair and can be interpreted as a relative energy-momentum density. It is distinguished as a geometric object by its property of being invariant under suitable diffeomorphisms fixing the boundary, and that act on (either) one of the argument metrics. Specialising to the case of an ALH metric $h$ that is suitably asymptotically related to a locally hyperbolic conformally compact metric, we show that the cocycle determines an absolute invariant $c(h)$, which still is local in nature. This tractor-valued $(n-1)$-form field on the conformal infinity is canonically associated to $h$ (i.e. is not dependent on other choices) and is equivariant under the appropriate diffeomorphisms. Finally specialising further to the case that the boundary is a sphere and that a metric $h$ is asymptotically related to a hyperbolic metric on the interior, we show that the invariant $c(h)$ can be integrated over the boundary. The result pairs with solutions of the KID (Killing initial data) equation to recover the known description of hyperbolic mass integrals of Wang, and Chruściel--Herzlich.

A relative mass cocycle and the mass of asymptotically hyperbolic manifolds

Abstract

We construct a cocycle that, for a given -manifold, maps pairs of asymptotically locally hyperbolic (ALH) metrics to a tractor-valued -form field on the conformal infinity. This requires the metrics to be asymptotically related to a given order that depends on the dimension. It then provides a local geometric quantity on the boundary that is naturally associated to the pair and can be interpreted as a relative energy-momentum density. It is distinguished as a geometric object by its property of being invariant under suitable diffeomorphisms fixing the boundary, and that act on (either) one of the argument metrics. Specialising to the case of an ALH metric that is suitably asymptotically related to a locally hyperbolic conformally compact metric, we show that the cocycle determines an absolute invariant , which still is local in nature. This tractor-valued -form field on the conformal infinity is canonically associated to (i.e. is not dependent on other choices) and is equivariant under the appropriate diffeomorphisms. Finally specialising further to the case that the boundary is a sphere and that a metric is asymptotically related to a hyperbolic metric on the interior, we show that the invariant can be integrated over the boundary. The result pairs with solutions of the KID (Killing initial data) equation to recover the known description of hyperbolic mass integrals of Wang, and Chruściel--Herzlich.

Paper Structure

This paper contains 18 sections, 18 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Setup and tractors
  3. Conformally compact metrics
  4. ALH-metrics and adapted defining functions
  5. The basic setup
  6. Tractors
  7. Boundary values of tractors
  8. The scale tractor
  9. The tractor mass cocycle
  10. The contribution from the trace
  11. The contribution from the trace-free part
  12. Diffeomorphisms
  13. Diffeomorphisms asymptotic to the identity
  14. The relation to adapted defining functions
  15. Aligned metrics
  16. ...and 3 more sections

Key Result

Proposition 2.1

Let $\overline{M}$ be a smooth manifold with boundary $\partial M$ and interior $M$, which is endowed with a conformal structure $c$. Then a metric $g$ on $M$ which lies in $c|_M$ is conformally compact if an only if any non-zero section $\sigma\in\Gamma(\mathcal{E}[1]|_M)$, which is parallel for th

Theorems & Definitions (40)

  • Proposition 2.1
  • Definition 2.2
  • Remark 2.3
  • Proposition 2.4
  • Proposition 2.5
  • Proposition 2.6
  • proof
  • Proposition 3.1
  • proof
  • Corollary 3.2
  • ...and 30 more