It was recently shown by R. Souam and E. Toubiana [33] that the (non constantly curved) Berger spheres do not contain totally umbilic surfaces. Nevertheless in this article we show, by perturbative arguments, that all analytic metrics su ciently close to the round metric g0 on S possess generalized totally umbilic 2-spheres, namely critical points of the conformal Willmore functional ∫ Σ |A◦|2 ...

We answer the following question: Let λ, μ, ν be arbitrary real numbers. Does there exist a 3-dimensional homogeneous Riemannian manifold whose eigenvalues of the Ricci tensor are just λ, μ and ν ? It is essential to observe that inspite of the fact that the curvature of a 3–manifold is uniquely determined by the Ricci tensor, nevertheless, the constancy of all its three eigenvalues does not im...

Gravitational instantons, solutions to the euclidean Einstein equations, with topology R3XS1 arise naturally in any discussion of finite temperature quantum gravity. This Letter shows that all such instantons (irrespective of their interior behaviour) must have the same asymptotic structure as the Schwarzschild instanton. Using this, it can be shown that if the Ricci tensor of the manifold is n...