The Bernstein problem for Sobolev intrinsic graphs in the Heisenberg group

TL;DR

This paper proves that stable Sobolev intrinsic graphs in the first Heisenberg group are necessarily intrinsic planes, extending previous results beyond Lipschitz regularity under certain integrability conditions.

Contribution

It extends the Bernstein problem to Sobolev intrinsic graphs in the Heisenberg group, showing stability implies flatness under weaker regularity assumptions.

Findings

Stable Sobolev intrinsic graphs are intrinsic planes.

Extension of Bernstein problem beyond Lipschitz class.

Results under appropriate derivative integrability conditions.

Abstract

In the first Heisenberg group, we study entire, locally Sobolev intrinsic graphs that are stable for the sub-Riemannian area. We show that, under appropriate integrability conditions for the derivatives, the intrinsic graph must be an intrinsic plane, i.e., a coset of a two dimensional subgroup. This result extends \cite{arXiv:1809.04586} beyond the Lipschitz class.