The Ricci flow of asymptotically hyperbolic mass and applications

T. Balehowsky, E. Woolgar

Forskningsoutput: TidskriftsbidragArtikelVetenskapligPeer review

Sammanfattning

We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbolic manifold of dimension n>2 decays smoothly to zero exponentially in the flow time. From this, we obtain a no-breathers theorem and a Ricci flow based, modified proof of the scalar curvature rigidity of zero-mass asymptotically hyperbolic manifolds. We argue that the nonconstant time evolution of the asymptotically hyperbolic mass is natural in light of a conjecture of Horowitz and Myers, and is a test of that conjecture. Finally, we use a simple parabolic scaling argument to produce a heuristic "derivation" of the constancy of ADM mass under asymptotically flat Ricci flow, starting from our decay formula for the asymptotically hyperbolic mass under the curvature-normalized flow.
Originalspråkengelska
TidskriftJournal of Mathematical Physics
Volym53
ISSN0022-2488
DOI
StatusPublicerad - 12 jul 2012
Externt publiceradJa
MoE-publikationstypA1 Tidskriftsartikel-refererad

Bibliografisk information

Revised in accord with referee comments, typos and minor errors corrected, Appendix B added, footnotes in-lined; version accepted for publication

Vetenskapsgrenar

  • 111 Matematik

Citera det här

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The Ricci flow of asymptotically hyperbolic mass and applications. / Balehowsky, T.; Woolgar, E.

I: Journal of Mathematical Physics, Vol. 53, 12.07.2012.

Forskningsoutput: TidskriftsbidragArtikelVetenskapligPeer review

TY - JOUR

T1 - The Ricci flow of asymptotically hyperbolic mass and applications

AU - Balehowsky, T.

AU - Woolgar, E.

N1 - Revised in accord with referee comments, typos and minor errors corrected, Appendix B added, footnotes in-lined; version accepted for publication

PY - 2012/7/12

Y1 - 2012/7/12

N2 - We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbolic manifold of dimension n>2 decays smoothly to zero exponentially in the flow time. From this, we obtain a no-breathers theorem and a Ricci flow based, modified proof of the scalar curvature rigidity of zero-mass asymptotically hyperbolic manifolds. We argue that the nonconstant time evolution of the asymptotically hyperbolic mass is natural in light of a conjecture of Horowitz and Myers, and is a test of that conjecture. Finally, we use a simple parabolic scaling argument to produce a heuristic "derivation" of the constancy of ADM mass under asymptotically flat Ricci flow, starting from our decay formula for the asymptotically hyperbolic mass under the curvature-normalized flow.

AB - We consider the evolution of the asymptotically hyperbolic mass under the curvature-normalized Ricci flow of asymptotically hyperbolic, conformally compactifiable manifolds. In contrast to asymptotically flat manifolds, for which ADM mass is constant during Ricci flow, we show that the mass of an asymptotically hyperbolic manifold of dimension n>2 decays smoothly to zero exponentially in the flow time. From this, we obtain a no-breathers theorem and a Ricci flow based, modified proof of the scalar curvature rigidity of zero-mass asymptotically hyperbolic manifolds. We argue that the nonconstant time evolution of the asymptotically hyperbolic mass is natural in light of a conjecture of Horowitz and Myers, and is a test of that conjecture. Finally, we use a simple parabolic scaling argument to produce a heuristic "derivation" of the constancy of ADM mass under asymptotically flat Ricci flow, starting from our decay formula for the asymptotically hyperbolic mass under the curvature-normalized flow.

KW - math.DG

KW - gr-qc

KW - hep-th

KW - 111 Mathematics

U2 - 10.1063/1.4732118

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M3 - Article

VL - 53

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

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