Ricci flow of homogeneous manifolds and its solitons

Riemannian Geometry

Soit (M,g ij (x)) une variete de Riemann a n dimensions complete non compacte de tenseur de complexe riemannien {R ijkl } satisfaisant: |R ijkl | 2 ≤k 0 sur M, ou 0 0 dependant seulement de n et de k 0 telle que l'equation d'evolution δg ij (x,t)|δt=−2R ij (x,t) sur M, g ij (x,o)=g ij (x) ∀x∈M ait une solution lisse g ij (x,t)>0 pour un temps court 0≤t≤T(u,k 0 ), et satisfait l'estimation suivante: pour un entier m≥0, il existe des constantes C m >0 dependant seulement de n, m et k 0 telles que sup x∈M |v m R ijkl (x,t)| 2 ≤C m |t m , 0≤t≤T(u,k 0 )

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Deforming the metric on complete Riemannian manifolds

Naturally reductive metrics and Einstein metrics on compact Lie groups

/pdf/a-variational-approach-for-compact-homogeneous-einstein-w4lwi0qggi.pdf

A variational approach for compact homogeneous Einstein manifolds

The aim of this paper is to study the problem of which solvable Lie groups admit an Einstein left invariant metric. The space \({\mathcal{N}}\) of all nilpotent Lie brackets on \({\mathbb{R}^n}\) parametrizes a set of (n + 1)-dimensional rank-one solvmanifolds \({\{S_{\mu}:\mu\in\mathcal{N}\}}\), containing the set of all those which are Einstein in that dimension. The moment map for the natural GL n -action on \({\mathcal{N}}\), evaluated at \({\mu\in\mathcal{N}}\), encodes geometric information on S μ and suggests the use of strong results from geometric invariant theory. For instance, the functional on \({\mathcal{N}}\) whose critical points are precisely the Einstein S μ ’s, is the square norm of this moment map. We use a GL n -invariant stratification for the space \({\mathcal{N}}\) and show that there is a strong interplay between the strata and the Einstein condition on the solvmanifolds S μ . As an application, we obtain criteria to decide whether a given nilpotent Lie algebra can be the nilradical of a rank-one Einstein solvmanifold or not. We find several examples of \({\mathbb{N}}\)-graded (even 2-step) nilpotent Lie algebras which are not. A classification in the 7-dimensional, 6-step case and an existence result for certain 2-step algebras associated to graphs are also given.

Ricci flow of homogeneous manifolds and its solitons

References

Riemannian Geometry

Deforming the metric on complete Riemannian manifolds

Naturally reductive metrics and Einstein metrics on compact Lie groups

A variational approach for compact homogeneous Einstein manifolds

Einstein solvmanifolds: existence and non-existence questions

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