Three-dimensional graph surfaces on five-dimensional Carnot-Carathéodory spaces
type: Item
instance of: scholarly article
title: Three-dimensional graph surfaces on five-dimensional Carnot-Carathéodory spaces
zbMATH Open document ID: 1412.53052
DOI: 10.1134/S0037446618040080
author: M. Karmanova
published in: Siberian Mathematical Journal
publication date: 2018-11-13T00:00:00Z
review text: 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.
reviewed by: Nathaniel Eldredge
Mathematics Subject Classification ID: 53C17
Mathematics Subject Classification ID: 49Q05
Mathematics Subject Classification ID: 53A10
zbMATH DE Number: 6976643
zbMATH Keywords: Carnot-Carathéodory space
zbMATH Keywords: Carnot manifold
zbMATH Keywords: graph mapping
zbMATH Keywords: polynomial sub-Riemannian differentiability
zbMATH Keywords: area formula
zbMATH Keywords: minimal surface
MaRDI profile type: Publication
cites work: Graph surfaces over three-dimensional Lie groups with sub-Riemannian structure
cites work: The graphs of Lipschitz functions and minimal surfaces on Carnot groups
cites work: Maximal surfaces on five-dimensional group structures
cites work: Q3517273
cites work: Graph surfaces on five-dimensional sub-Lorentzian structures
cites work: Sub-Riemannian structures on 3D Lie groups
cites work: Q4676076
cites work: Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model
cites work: Q3172397
cites work: Q5459459
cites work: Regular hypersurfaces, intrinsic perimeter and implicit function theorem in Carnot groups
cites work: Parametrizations of level surfaces of real-valued mappings of Carnot groups
cites work: One-Dimensional Level Sets of hc-Differentiable Mappings of Carnot–Carathéodory Spaces
cites work: Approximate differentiability of mappings of Carnot-Carathéodory spaces
cites work: Q4718347
cites work: Balls and metrics defined by vector fields. I: Basic properties
cites work: Geometry of Carnot-Carathéodory Spaces, Differentiability, Coarea and Area Formulas
cites work: On Local Approximation Theorem on Equiregular Carnot-Carathéodory Spaces
cites work: Stratified Lie Groups and Potential Theory for their Sub-Laplacians
cites work: Fine properties of basis vector fields on Carnot-Carathéodory spaces under minimal assumptions on smoothness
cites work: Set functions and their applications in the theory of Lebesgue and Sobolev spaces. I
cites work: Q5715075
cites work: Variations of nonholonomic-valued mappings and their applications to maximal surface theory
cites work: Maximal graph surfaces on four-dimensional two-step sub-Lorentzian structures
full work available at URL: s0037446618040080
OpenAlex ID: W2891117040
description: scientific article; zbMATH DE number 6976643