The cut locus of a C^2 surface in the Heisenberg group

My master thesis deals with some fine properties of the natural Carnot-Carathéodory distance in the Heisenberg group. In particular, we are interested in properties related to the cut locus of a smooth surface (for us, the term smooth indicates a second order differentiability). The cut locus of a closed subset S of the n-dimensional Euclidean space, denoted by cut(S), is the set containing the endpoints of maximal segments that minimize the distance to S. In the Euclidean case many properties of the cut locus are well known if S is the smooth boundary of an open set. For example the cut loci of such surfaces are closed (this is the fact we will be most interested in). Such results are still valid if the Euclidean metric is replaced with any smooth Riemannian metric. Not all the known properties of the cut locus in Riemannian geometry are also known to hold in the sub-Riemannian case. So our goal is to generalize and prove some of them in the Heisenberg group, which has the simplest sub-Riemannian structure. In particular we are very interested in proving that the cut loci of surfaces which are the smooth boundary of open sets in the Heisenberg group are closed sets. At the moment we are not able to give a proof of the closure of the cut locus. So we are looking for new properties of the Carnot distance which may be related with the cut locus of a smooth surface in the Heisenberg group and that may be useful to prove its closure. Precisely, we are investigating in points conjugate to the surface S, since they are strictly connected with the cut locus in the Riemannian case. Just about conjugate points, we will show some new small results at the end of this work.