Stability of hyperbolic space under Ricci flow

Abstract

We study the Ricci flow of initial metrics which are C 0 -perturbations of the hyperbolic metric on H n . If the perturbation is bounded in the L 2 -sense, and small enough in the C 0 -sense, then we show the following: In dimensions four and higher, the scaled Ricci harmonic map heat flow of such a metric converges smoothly, uniformly and exponentially fast in all C k -norms and in the L 2 -norm to the hyperbolic metric as time approaches infinity. We also prove a related result for the Ricci flow and for the two-dimensional conformal Ricci flow.

On the Uniqueness of Ricci Flow

“…1 from a sufficiently small perturbation of the Euclidean metric on R n and [25] for a similar result in hyperbolic space. In [32], Xu constructed a solution from a metric g 0 which satisfies a Sobolev inequality and the curvature is bounded in some L p sense.…”

The Kähler–Ricci flow around complete bounded curvature Kähler metrics

“…As another application, we have the following stability result for (1.1) around complex spaceforms. In the Riemannian case, the stability of spaceforms was first studied in [23,24]. We show that in the Kähler category, we do not require the ǫ-fairness required in those works.…”

Volume Comparison of Conformally Compact Manifolds with Scalar Curvature R ≥ −n (n − 1)

“…where the components of V are given by V i := g ij V j , then we obtain a family of smooth diffeomorphisms Φ t for t > 0 such that if g(t), t ∈ [0, T ) is a solution to NRDF (2),ḡ(t) := Φ * t g(t), t ∈ [0, T ) is a solution to NRF (1). There are several papers which investigated the stability of hyperbolic space under NRF [14,20,2,3]. In [20], Schnürer, Schulze and Simon used the NRDF to get the stability of hyperbolic space.…”

“…There are several papers which investigated the stability of hyperbolic space under NRF [14,20,2,3]. In [20], Schnürer, Schulze and Simon used the NRDF to get the stability of hyperbolic space. To be precise, under the assumptions that g − h C 0 (H n ,h) ≤ ǫ and g − h L 2 (H n ,h) ≤ K, the NRDF starting from g with background metric being hyperbolic metric h exists globally.…”