H-minimal graphs of low regularity in the Heisenberg group

TL;DR

This paper studies low-regularity H-minimal graphs in the Heisenberg group, revealing their geometric structure, methods of constructing piecewise solutions, and conditions affecting the smoothness of solutions to the Dirichlet problem.

Contribution

It provides new insights into the structure of low-regularity H-minimal graphs and explores conditions for smooth solutions to the Dirichlet problem in the Heisenberg group.

Findings

H-minimal graphs with certain regularity are ruled surfaces with C^2 seed curves.

Conditions for the existence of smooth solutions to the Dirichlet problem are identified.

Examples show smooth data may not produce smooth solutions, illustrating limitations of regularity.

Abstract

In this paper we investigate H-minimal graphs of lower regularity. We show that noncharactersitic C^1 H-minimal graphs whose components of the unit horizontal Gauss map are in W^{1,1} are ruled surfaces with C^2 seed curves. In a different direction, we investigate ways in which patches of C^1 H-minimal graphs can be glued together to form continuous piecewise C^1 H-minimal surfaces. We apply these description of H-minimal graphs to the question of the existence of smooth solutions to the Dirichlet problem with smooth data. We find a necessary condition for the existence of smooth solutions and produce examples where the conditions are satisfied and where they fail. In particular we illustrate the failure of the smoothness of the data to force smoothness of the solution to the Dirichlet problem by producing a class of smooth curves whoses H-minimal spanning graphs cannot be C^2.