Intrinsic Regular Surfaces of low codimension in Heisenberg groups

Abstract

In this paper we study intrinsic regular submanifolds of $\mathbb{H}^n$, of low co-dimension in relation with the regularity of their intrinsic parametrization. We extend some results proved for one co-dimensional $\mathbb{H}$-regular surfaces, characterizing uniformly intrinsic differentiable functions $\phi$ acting between two complementary subgroups of the Heisenberg group $\mathbb{H}^n$, with target space horizontal of dimension $k$, with $1 \leq k \leq n$, in terms of the Euclidean regularity of its components with respect to a family of non linear vector fields $\nabla^{\phi_j}$. Moreover, we show how the area of the intrinsic graph of $\phi$ can be computed through the component of the matrix identifying the intrinsic differential of $\phi$.