The Bernstein Problem in the Heisenberg Group

TL;DR

This paper proves a Bernstein-type theorem for minimal surfaces in the first Heisenberg group, characterizing entire minimal graphs as either vertical planes or surfaces with constant curvature seed curves.

Contribution

It extends Bernstein's classical result to the sub-Riemannian setting of the Heisenberg group, identifying conditions for minimal surfaces to be planes or have specific curvature properties.

Findings

H-minimal surfaces that are graphs over a plane are either vertical planes or have seed curves with constant curvature.

The theorem characterizes entire minimal graphs in the Heisenberg group.

Provides a classification of C^2 connected minimal surfaces in the Heisenberg group.

Abstract

We establish the following theorem of Bernstein type for the first Heisenberg group: Let S be a C^2 connected H-minimal surface which is a graph over some plane P, then S is either a non-characteristic vertical plane, or its generalized seed curve satisfies a type of constant curvature condition.