The Bernstein problem for $(X,Y)$-Lipschitz surfaces in three-dimensional sub-Finsler Heisenberg groups

TL;DR

This paper proves that in the sub-Finsler Heisenberg group, complete, oriented, connected, and stable Lipschitz surfaces are necessarily vertical planes, extending classical Bernstein problem results to this geometric setting.

Contribution

It establishes a Bernstein-type theorem for Lipschitz surfaces in sub-Finsler Heisenberg groups, characterizing stable solutions as vertical planes.

Findings

Stable Lipschitz surfaces are vertical planes in the sub-Finsler Heisenberg group.

The result applies to entire intrinsic graphs of Euclidean Lipschitz functions.

The theorem extends classical Bernstein results to a sub-Finsler geometric context.

Abstract

We prove that in the Heisenberg group $\mathbb{H}^1$ with a sub-Finsler structure, an $(X,Y)$-Lipschitz surface which is complete, oriented, connected and stable must be a vertical plane. In particular, the result holds for entire intrinsic graphs of Euclidean Lipschitz functions.