The Ricci flow of asymptotically hyperbolic mass and applications

TL;DR

This paper studies how the asymptotically hyperbolic mass evolves under Ricci flow, showing it decays exponentially to zero, and applies this to prove rigidity results and explore connections with the ADM mass in flat cases.

Contribution

It demonstrates the decay of asymptotically hyperbolic mass under Ricci flow and provides a new proof of scalar curvature rigidity, linking mass evolution to geometric and physical conjectures.

Findings

Mass decays exponentially under Ricci flow

No-breathers theorem established for asymptotically hyperbolic manifolds

Heuristic derivation of ADM mass constancy in flat case

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.