Optimal regularity for Bernstein rigidity in the first Heisenberg group

Determine the weakest regularity assumption that guarantees the rigidity theorem for complete, two-sided, connected, stable, non-characteristic hypersurfaces in the first Heisenberg group, namely, that such hypersurfaces are vertical planes.

Background

The paper reviews a sequence of results establishing that sufficiently regular, complete, stable, non-characteristic hypersurfaces in the first Heisenberg group must be vertical planes. The required regularity has progressively been weakened from smooth and $C^2$ assumptions to $C^1$, Euclidean Lipschitz, and Euclidean Sobolev regularity.

The authors explicitly identify the remaining question as determining the best possible regularity threshold under which the Bernstein-type rigidity conclusion continues to hold. This problem is unresolved and is cited to prior work.