It is well known that, in the case of finite area hyperbolic surfaces, the length sprectrum (the collection of length of all closed geodesics) and the spectrum of the laplacian (acting on functions) are the same. See for instance :
http://www.mathcs.emory.edu/~davidb/spths.pdf
for an introduction. A quasi-fuchsian manifold is a threefold isometric to the quotient of $\mathbb{H}^3$ by a representation of a surface group in $SL_2(\mathbb{C})$ which is in the same component of representations in $PSL_2 (\mathbb{R}) \subset PSL_2(\mathbb{C})$.
This kind of manifolds have infinite volume, but a large part of dynamical data are contained in the convex hull of such manifold (which is compact), in particular all closed geodesics belongs to the convex hull. Therefore my question is :
Is there exist a link between the length spectrum of a quasi-fuchsian manifold and the laplacian one but in restriction to the convex hull (as a Neumann or Dirichlet problem or something else) ?