Intrinsic regular surfaces of low co-dimension in Heisenberg groups – Francesca Corni (Universita’ di Bologna)

Venue

Sala Seminari (Dip. Matematica).

Abstract

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-dimensionkcan be consideredG-regular surfaces (H-regular ifG=Hn), i.e. level sets of continuously Pansu- differentiable functionsf:G−→Rkwhose differential is subjective. If it is possible to split the groupGin the product of two suitable homogeneous complementary subgroupsMandH, aG-regular surfaces can be locally seen as anuniformly intrinsic differentiable graphs, defined by a unique continuous functionφacting betweenMandH. Moreover, it turns out that any one co-dimensionalH-regular surface locally defines an implicit functionφ, which is of classC1with respect to a suitable non linear vector field∇φexpressed in terms of the functionφitself. We extend some of these results characterizing uniformly intrinsic differentiable functionsφacting between two complementary subgroups with target space horizontal of dimensionk, 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 ofφcan be computed through the component of the matrix identifying the intrinsic differential ofφ.

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