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Stability and instability of Einstein manifolds with prescribed asymptotic geometry

The Ricci flow is a geometric evolution equation that plays a fundamental role in modern Riemannian geometry.

To get a better understanding of the long time behaviour of the Ricci flow and of its singularities, it is important to study the stability of its stationary points on the space of metrics modulo homotheties, which are called Ricci solitons.

While the connection between linear stability, integrability and dynamical stability is now well understood in the compact case, the noncompact situation is understood only for particular examples or under very strict conditions. A general criterion for dynamical instability has not been shown so far. The aim of this project is to improve the knowledge about these questions in the noncompact case.

 


Publications

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.

 

JournalCalc. Var. Part. Differ. Eq.
PublisherSpringer
Volume58
Pages75
Link to preprint version
Link to published version

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry23Spectral geometry, index theory and geometric flows on singular spaces

We consider the long-time behaviour of the mean curvature flow of spacelike hypersurfaces in the Lorentzian product manifold M×R, where M is asymptotically flat. If the initial hypersurface F0⊂M×R is uniformly spacelike and asymptotic to M×{s} for some s∈R at infinity, we show that the mean curvature flow starting at F0 exists for all times and converges uniformly to M×{s} as t→∞.

 

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry29Curvature flows without singularities30Nonlinear evolution equations on singular manifolds31Solutions to Ricci flow whose scalar curvature is bounded in Lp.

In this paper we discuss Perelman's Lambda-functional, Perelman's Ricci shrinker entropy as well as the Ricci expander entropy on a class of manifolds with isolated conical singularities. On such manifolds, a singular Ricci de Turck flow preserving the isolated conical singularities exists by our previous work. We prove that the entropies are monotone along the singular Ricci de Turck flow. We employ these entropies to show that in the singular setting, Ricci solitons are gradient and that steady or expanding Ricci solitons are Einstein.

 

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry30Nonlinear evolution equations on singular manifolds

We prove that if an ALE Ricci-flat manifold (M,g) is linearly stable and integrable, it is dynamically stable under Ricci flow, i.e. any Ricci flow starting close to g exists for all time and converges modulo diffeomorphism to an ALE Ricci-flat metric close to g. By adapting Tian's approach in the closed case, we show that integrability holds for ALE Calabi-Yau manifolds which implies that they are dynamically stable.

 

Related project(s):
21Stability and instability of Einstein manifolds with prescribed asymptotic geometry

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Team Members

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