The Ricci flow of asymptotically hyperbolic mass and applications

Abstract

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.

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

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Cited by in corpus (5)

• The rigid Horowitz-Myers conjecture
• Reduction Arguments for Geometric Inequalities Associated With Asymptotically Hyperboloidal Slices
• Constructing Lifshitz spaces using the Ricci flow
• Static flow on complete noncompact manifolds I: short-time existence and asymptotic expansions at conformal infinity
• The Higgs mechanism and geometrical flows for two-manifolds