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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

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