Asymptotically hyperbolic normalized Ricci flow and rotational symmetry

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.

Convergence stability for Ricci flow on manifolds with bounded geometry

“…This is not perturbative since g 0 may be far away from the hyperbolic metric, and in particular might have some positive sectional curvatures. The new result here asserts the stability of the entire trajectory of such solutions, in the sense that if g 1 is any not necessarily rotationally symmetric AH metric sufficiently close to a rotationally symmetric metric g 0 satisfying the condition of [5], then the Ricci flow solution g 1 (t) starting at g 1 also converges to the standard hyperbolic metric. This is an instance of convergence stability, as introduced in [3].…”

“…Suppose that g is a smooth, rotationally symmetric AH metric on B n such that g − g h ∈ C k,α µ (M ) for some k ≥ 2 and µ ∈ (0, n − 1), but where this norm is only assumed to be finite, but not small. The main theorem of [5] asserts that if the 'tangential' sectional curvatures of g, i.e., the sectional curvatures associated to 2-planes tangent to the angular SO(n) orbits, are strictly negative, then the Ricci flow g(t) emanating from g exists for all time, remains AH and rotationally symmetric, and converges exponentially to g h . The novelty, of course, is that g = g(0) may be quite far from g h .…”

“…The second topic is a new stability result about Ricci flow of complete, asymptotically hyperbolic (AH) metrics. The starting point is a result by the first author and Woolgar [5], which states that if g 0 is AH and rotationally symmetric, and satisfies a mild geometric hypothesis to be described below, then the Ricci flow solution g(t) starting at g 0 converges to the standard hyperbolic metric. This is not perturbative since g 0 may be far away from the hyperbolic metric, and in particular might have some positive sectional curvatures.…”

“…The proof relies on well-posedness of the Ricci-DeTurck flow. In Section 3 we review the Ricci flow convergence result for asymptotically hyperbolic rotationally symmetric metrics from [5] and based on this, prove convergence stability for Ricci flow near any such trajectory.…”