Entropy, Critical Exponent and Immersed Surfaces in Hyperbolic 3-Manifolds

We consider a $\pi_{1}$--injective immersion $f:\Sigma\to M$ from a compact surface $\Sigma$ to a hyperbolic 3--manifold $M$. Let $\Gamma$ denote the copy of $\pi_{1}\Sigma$ in $\mathrm{Isom}({\mathbb{H}}^{3})$ induced by the immersion and $\delta(\Gamma)$ be the critical exponent. Suppose $\Gamma$ is convex cocompact and $\Sigma$ is negatively curved, we prove that there are two geometric constants $C_{1}(\Sigma,M)$ and $C_{2}(\Sigma,M)$ not bigger than $1$ such that $C_{1}(\Sigma,M)\cdot\delta_{\Gamma}\leq h(\Sigma)\leq C_{2}(\Sigma,M)\cdot\delta_{\Gamma}$, where $h(\Sigma)$ is the topological entropy of the geodesic flow on. When $f$ is an embedding, we show that $C_{1}(\Sigma,M)$ and $C_{2}(\Sigma,M)$ are exactly the geodesic stretches (a.k.a. Thurston's intersection number) with respect to certain Gibbs measures. Moreover, we prove the rigidity phenomenon arising from this inequality. Lastly, as an application, we discuss immersed minimal surfaces in hyperbolic 3--manifolds and these discussions lead us to results similar to A. Sanders' work on the moduli space of $\Sigma$ introduced by C. Taubes.