Talks

University of Padova

Title: Tubular neighborhoods in sub-Riemannian geometry: Steiner's and Weyl's tube formulae

Abstract: Steiner and Weyl proved that the volume of the tubular neighborhood of a submanifold in Rn is a polynomial of degree n in the "size" of the tube. 

The coefficients of such a polynomial carry information about the curvature of the submanifold. 

In this talk, we investigate the validity of a Steiner and Weyl-like formula where the ambient space is a sub-Riemannian manifold, extending previous results obtained in the Heisenberg group. 

Furthermore, in the case of a 3D contact sub-Riemannian manifold, we provide a geometric interpretation of the coefficients of the Taylor expansion of the volume as the size of the tube tends to zero in terms of sub-Riemannian curvature objects.

University of Fribourg

Title: Metabelian distributions and sub-Riemannian geodesics.

Abstract: This talk focuses on those particular sub-Riemannian structures whose distribution is metabelian. On the one hand, we discuss some results on the integrability of the normal Hamiltonian flow and on globally length-minimizing curves. 

On the other hand, we discuss abnormal curves. We prove that their projection to some lower dimensional manifold must stay inside an analytic variety. As a consequence, for rank-2 metabelian distributions, geodesics are continuously differentiable. This talk is based on joint works with Alejandro Bravo-Doddoli, Enrico Le Donne and Alessandro Socionovo.

University of Trento

Title: The Bernstein problem in sub-Riemannian Heisenberg groups: origins and new developments.

Abstract: The so-called Bernstein problem consists in characterizing global minimizers of suitable perimeter functionals. Roughly speaking, is it true that (smooth) boundaries of global perimeter minimizers are flat in a certain sense? While this topic is well understood in the Euclidean framework, the Bernstein problem in sub-Riemannian Heisenberg groups leaves many interesting questions still unanswered, especially with regard to its high-dimensional formulation. A crucial step in the study of minimal surfaces in  H^1, where the Bernstein problem is answered affirmatively, is to show that they are ruled by horizontal lines. In this seminar, after a survey of the known results, we introduce a generalization of the notion of ruled surface to higher dimensional Heisenberg groups. We link this merely differential notion with the vanishing of a suitable horizontal second fundamental form, and we provide a characterization of those hypersurfaces which share the above-mentioned properties. To conclude, we discuss some consequences and we present some possible developments.