Intrinsic regular surfaces of low codimension in Heisenberg groups

Abstract

In this paper we study intrinsic regular submanifolds of ${\mathbf{H}}^n$ of low codimension in relation with the regularity of their intrinsic parametrization. We extend some results proved for $\mathbf{H}$-regular surfaces of codimension 1 to $\mathbf{H}$-regular surfaces of codimension $k$, with $1\le k\le n$. We characterize uniformly intrinsic differentiable functions, $\varphi$, acting between two complementary subgroups of the Heisenberg group ${\mathbf{H}}^n$, with target space horizontal of dimension $k$, in terms of the Euclidean regularity of their components with respect to a family of non linear vector fields ${\mathrm{\nabla }}^{{\varphi }_j}$. Moreover, we show how the area of the intrinsic graph of $\varphi$ can be computed in terms of the components of the matrix representing the intrinsic differential of $\varphi$.