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 \(1 \leq k \leq n\). We characterize uniformly intrinsic differentiable functions, \(\phi\), 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 \(\nabla^{\phi_j}\). Moreover, we show how the area of the intrinsic graph of \(\phi\) can be computed in terms of the components of the matrix representing the intrinsic differential of \(\phi\).