Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

TL;DR

This paper studies the normalized Ricci flow on asymptotically hyperbolic, conformally compactifiable manifolds, proving long-term existence and convergence to hyperbolic space under certain curvature conditions.

Contribution

It establishes existence, uniqueness, and convergence results for the normalized Ricci flow on asymptotically hyperbolic manifolds with rotational symmetry, extending understanding of geometric flows in this setting.

Findings

Flow exists up to curvature blow-up time

Flow converges exponentially to hyperbolic space under negative sectional curvature

Initial curvature conditions prevent minimal hypersphere formation

Abstract

We consider the normalized Ricci flow evolving from an initial metric which is conformally compactifiable and asymptotically hyperbolic. We show that there is a unique evolving metric which remains in this class, and that the flow exists up to the time where the norm of the Riemann tensor diverges. Restricting to initial metrics which belong to this class and are rotationally symmetric, we prove that if the sectional curvature in planes tangent to the orbits of symmetry is initially nonpositive, the flow starting from such an initial metric exists for all time. Moreover, if the sectional curvature in planes tangent to these orbits is initially negative, the flow converges at an exponential rate to standard hyperbolic space. This restriction on sectional curvature automatically rules out initial data admitting a minimal hypersphere.