On sub-Riemannian caustics and wave fronts for contact distributions in the three space

In a number of previous papers of the first and third authors, caustics, cut-loci, spheres, and wave fronts of a system of sub-Riemannian geodesics emanating from a point q0 were studied. It turns out that only certain special arrangements of classical Lagrangian and Legendrian singularities occur outside q0. As a consequence of this, for instance, the generic caustic is a globally stable object outside the origin q0. Here we solve two remaining stability problems. The first part of the paper shows that in fact generic caustics have moduli at the origin, and the first module that occurs has a simple geometric interpretation. On the contrary, the second part of the paper shows a stability result at q0. We define the “big wave front”: it is the graph of the multivalued function arclength → wave-front reparametrized in a certain way. This object is a three-dimensional surface that also has a natural structure of the wave front. The projection of the singular set of this “big wave front” on the 3-dimensional space is nothing else but the caustic. We show that in fact this big wave front is Legendre-stable at the origin.