  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.
