Stability of Ricci de Turck flow on singular spaces

Abstract

In this paper we establish stability of the Ricci de Turck flow near Ricci-flat metrics with isolated conical singularities. More precisely, we construct a Ricci de Turck flow which starts sufficiently close to a Ricci-flat metric with isolated conical singularities and converges to a singular Ricci-flat metric under an assumption of integrability, linear and tangential stability. We provide a characterization of conical singularities satisfying tangential stability and discuss examples where the integrability condition is satisfied.

Ricci de Turck flow on incomplete manifolds

“…We should point out that short-time existence and further properties of a Ricci de Turck flow on incomplete manifolds has already been established in the special case of manifolds with conical or more generally wedge singularities in varying dimensions in [MRS15], [BaVe14], [Ver16], [KrVe19a] and [Yin10], to name a few. These references deal with the flow that stays uniformly equivalent to the initial metric and hence preserves the initial singularity.…”

Ricci de Turck Flow on Singular Manifolds

“…Long time existence and stability of Ricci flow for small perturbations of Ricci flat metrics that are not flat, requires an integrability condition and other intricate geometric arguments. This has been the focus of the joint work with Kröncke [KrVe17].…”

“…certain simple Lie groups and rank-1 symmetric spaces of compact type. The actual statement in [KrVe17] also identifies the cases where tangential stability fails. Moreover [KrVe17] shows that the only example where (F, g F ) is weakly tangentially stable in the sense of Definition 8.1, but not tangentially stable is the case of a sphere.…”

“…The basic examples of spaces that are weakly tangentially stable but not tangentially stable are spaces with cross sections S f and RP f , or quotients of these. We refer to our work [KrVe17] for further details.…”

Spectra, rigidity and stability of sine-cones

“…For convenience, we call such singularties (strictly) tangentially stable. The topologies on the space of metrics in this subsection are induced by hybrid weighted Hölder norms H k,α γ for k ≥ 2,γ > 0 and α ∈ (0, 1), see [KV18,Definition 4.5] In [KV18], Boris Vertman and the author established a stability theorem for compact Ricci-flat conifolds under the singular Ricci-de Turck flow. The methods used in this paper can also be directly adapted to the stability of Einstein manifolds with isolated conical singularities under the volume normalized Ricci flow.…”