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Besson, Courtois and Gallot proved that on a compact manifold, a hyperbolic metric has minimal entropy among the metrics of same volume. Here we show that a hyperbolic metric is a saddle point for the bottom of spectrum of the universal cover. More precisely, the bottom of its spectrum is maximal among the conformal metrics of same volume, and minimal in its Ebin class.

Le bas du spectre d'une variété hyperbolique est un point selle