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  11. A Boundary-local Mass Cocycle And The Mass Of Asymptotically Hyperbolic Manifolds

A Boundary-Local Mass Cocycle and the Mass of Asymptotically Hyperbolic Manifolds

Andreas Čap1, A Rod Gover2

  • 1Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria.

Communications in Mathematical Physics
|September 18, 2024

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View abstract on PubMed

Summary
This summary is machine-generated.

We developed a new cocycle to define a relative energy-momentum density for asymptotically locally hyperbolic (ALH) manifolds. This geometric invariant is crucial for understanding the mass of hyperbolic manifolds.

Area of Science:

  • Differential Geometry
  • Mathematical Physics
  • Geometric Analysis

Background:

  • Asymptotically locally hyperbolic (ALH) manifolds are essential in general relativity and geometric analysis.
  • Understanding geometric invariants and boundary properties of these manifolds is a key challenge.
  • Existing methods for mass calculation in hyperbolic settings have limitations.

Purpose of the Study:

  • To construct a cocycle that maps pairs of ALH metrics to tractor-valued differential forms on the conformal infinity.
  • To define a local geometric quantity interpreted as relative energy-momentum density.
  • To derive an absolute invariant for ALH metrics and relate it to hyperbolic mass integrals.

Main Methods:

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  • Analysis of the cocycle's properties, including invariance under diffeomorphisms.
  • Specialization to conformally compact metrics and integration over spherical boundaries.
  • Connection to Killing initial data (KID) equations.

    • Main Results:

    • A tractor-valued differential form on conformal infinity is canonically associated with ALH metrics.
    • A local absolute invariant c(h) is derived for suitable ALH metrics.
    • Integration of c(h) over a spherical boundary recovers known hyperbolic mass integrals.
    • The invariant is shown to be local and equivariant under specific diffeomorphisms.

    Conclusions:

    • The constructed cocycle provides a powerful tool for studying ALH manifolds.
    • The derived invariant offers new insights into the geometric structure and mass of hyperbolic spaces.
    • This work bridges concepts in differential geometry and mathematical physics, with implications for general relativity.