In the first part of this talk, we present an elementary proof of the rigidity of the hyperbolic space as the unique Poincaré‐Einstein manifold whose boundary at infinity is the conformal round sphere. The proof relies on an inequality which relates the Yamabe invariant of the boundary with the one of a compactification of the bulk manifold. In a second part, we relate the Cheeger constant of such manifolds with the conformal type of the boundary at infinity . More precisely, we prove that this constant is exactly the dimension of the interior of the Poincaré‐Einstein manifold if and only if the Yamabe invariant of the boundary is non negative. Several applications are then discussed.
Simon Raulot (Frankreich): Yamabe invariants and Cheeger constant on Poincaré‐Einstein manifolds
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Foto: Barbara Mair