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Abstract: Hyperbolic manifolds are known to be strict extremum of riemanniann invariants as volume, entropy, bottom of the spectrum etc... I will talk of a differentiable rigidity of this property: if a riemannian n manifold does almost achieve an extremal value of such an invariant, is the underlying manifold diffeomorphic to an hyperbolic manifold? Analogue questions can be asked for representations. The fuschsian representations of a cocompact discrete subgroup of isometries of a hyperbolic space are known to be strict minimum of the Hausdorff dimension of the limit set among quasi-fuschsian representations. This property is not rigid in the real hyperbolic case and rigid in the complex hyperbolic case.
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