In classical work, Bowen and Margulis proved the equidistribution of closed geodesics in any closed hyperbolic manifold. Together with Jeremy Kahn and Vladimir Marković, we asked ourselves what happens in a 3-manifold if we replace curves by surfaces. (Slightly different aspects of this question have also been studied by Calegari, Marques and Neves.) The natural analogue of a closed geodesic is then a minimal surface, as totally geodesic surfaces exist only very rarely. Nevertheless, it still makes sense (for various reasons, in particular to ensure uniqueness of the minimal representative) to restrict our attention to surfaces that are almost totally geodesic.
The statistics of these surfaces then depend very strongly on how we order them: by genus, or by area. If we focus on surfaces whose area tends to infinity, we conjecture that they do indeed equidistribute; we proved a partial result in this direction. If, however, we focus on surfaces whose genus tends to infinity, the situation is completely opposite: we proved that they then accumulate onto the totally geodesic surfaces of the manifold (if there are any).
Fanny Kassel