Calculus of Variations and Geometric Measure Theory

Intrinsic regular surfaces of low co-dimension in Heisenberg groups

Francesca Corni

created by gelli on 22 Mar 2019

3 apr 2019 -- 17:00   [open in google calendar]

Sala Seminari (Dipartimento di Matematica di Pisa)


In Heisenberg groups, and, more in general, in Carnot groups, equipped with their Carnot- Carathodory metric, the analogous of regular (Euclidean) surfaces of low co-dimension k can be considered G-regular surfaces (H-regular if G = Hn), i.e. level sets of continuously Pansu- differentiable functions f : G −→ Rk whose differential is subjective. If it is possible to split the group G in the product of two suitable homogeneous complementary subgroups M and H, a G-regular surfaces can be locally seen as an uniformly intrinsic differentiable graphs, defined by a unique continuous function φ acting between M and H. Moreover, it turns out that any one co-dimensional H-regular surface locally defines an implicit function φ, which is of class C1 with respect to a suitable non linear vector field ∇φ expressed in terms of the function φ itself. We extend some of these results characterizing uniformly intrinsic differentiable functions φ acting between two complementary subgroups with target space horizontal of dimension k, with 1 ≤ k ≤ n, in terms of the Euclidean regularity of its components with respect to a family of non linear vector fields {∇φj }j=1,...,k. Eventually, we show how the area of the intrinsic graph of φ can be computed through the component of the matrix identifying the intrinsic differential of φ.