Entropy Rigidity, Pressure Metric, and Immersed Surfaces in Hyperbolic 3-Manifolds

Abstract

In this paper, we show an entropy rigidity result for immersed surfaces in hyperbolic 3-manifolds that relates dynamic and geometric quantities including entropy, critical exponent, and geodesic stretch. We then apply this result to \({\mathcal H}\) the minimal hyperbolic germs (a deformation space corresponding to the quasifuchsian space \(\mathcal {Q}\mathcal {F}\) proposed by Taubes). As a consequence, we recover the famous Bowen rigidity theorem for quasifuchsian representations. Moreover, we construct a Riemannian metric, i.e., the pressure metric, on the Fuchsian space \({\mathcal F}\subset \mathcal {H}\). We also discuss relations between the pressure metric, Sander’s metric, and Weil-Petersson metric on \({\mathcal F}\).