Three-dimensional graph surfaces on five-dimensional Carnot-Carathéodory spaces
Let $\mathbb{M}$ be a Carnot-Carathéodory space, which is a smooth manifold equipped with a frame of vector fields whose Lie brackets respect a certain grading. Such a space is called a Carnot manifold if the grading is actually a stratification. In this paper, we suppose we have a Carnot-Carathéodory space $\mathbb{M}$ and a Carnot manifold $\tilde{\mathbb{M}}$, both of dimension $3$, step $2$, and horizontal dimension $2$, embedded together in $\mathbb{R}^5$. These spaces carry a natural local group structure, and the Heisenberg group is used as a motivating example. Given a map $\varphi : D \subset \mathbb{M} \to \tilde{\mathbb{M}}$ where $D$ is open, the object of study in this paper is an appropriately defined \textit{graph mapping} $\varphi_{\Gamma} : D \to \mathbb{R}^5$ that produces a ``graph'' of the map $\varphi$ in a manner that respects the Carnot-Carathéodory structure. The paper includes results concerning: the regularity of the graph mapping $\varphi_{\Gamma}$; a formula for the intrinsic area of the image of $\varphi_{\Gamma}$; the regularity of the area functional with respect to $\varphi$; and conditions for a graph to have minimal area given a fixed boundary, including connections to a notion of sub-Riemannian mean curvature.
polynomial sub-Riemannian differentiability
scientific article; zbMATH DE number 6976643
scientific article