The Bernstein problem for Lipschitz intrinsic graphs in the Heisenberg group

TL;DR

This paper proves that in the Heisenberg group, Lipschitz intrinsic graphs with zero first variation and non-negative second variation are necessarily intrinsic planes, extending classical minimal surface results to a sub-Riemannian setting.

Contribution

It establishes a Bernstein-type theorem for Lipschitz intrinsic graphs in the Heisenberg group, characterizing minimal surfaces under certain variational conditions.

Findings

Intrinsic graphs with vanishing first variation and non-negative second variation are intrinsic planes.

Two examples demonstrate the sharpness of the main result.

The result extends classical Bernstein theorems to the sub-Riemannian context.

Abstract

We prove that, in the first Heisenberg group $\mathbb{H}$, an entire locally Lipschitz intrinsic graph admitting vanishing first variation of its sub-Riemannian area and non-negative second variation must be an intrinsic plane, i.e., a coset of a two dimensional subgroup of $\mathbb{H}$. Moreover two examples are given for stressing result's sharpness.