I study the statistical properties of quasifuchsian surfaces in a hyperbolic 3-manifold and how that relates to the structure of the manifold where they live. My PhD thesis is the following:
Limits of asymptotically fuchsian quasifuchsian surfaces in a closed hyperbolic 3-manifold (thesis) (arXiv)
Abstract: Let \(M\) be a closed hyperbolic 3-manifold. Let \(\nu_{Gr(M)}\) denote the probability volume (Haar) measure of the 2-plane Grassmannian \(Gr(M)\) of \(M\) and let \(\nu_T\) denote the area measure on \(Gr(M)\) of an immersed closed totally geodesic surface \(T\) in \(M\) We say a sequence of \(\pi_1\)-injective maps \(f_i:S_i\to M\) of surfaces \(S_i\) is asymptotically Fuchsian if \(f_i\) is \(K_i\)-quasifuchsian with \(K_i\to 1\) as \(i\to \infty\). We show that the set of weak-* limits of the probability area measures induced by \(f_i\) on \(Gr(M)\), where \(f_i:S_i\to M\) are increasingly Fuchsian minimal or pleated maps of closed connected surfaces \(S_i\), consists of all convex combinations of \(\nu_{Gr(M)}\) and the \(\nu_T\).