2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/64869We 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.62 pages, 9 figures. The first version of this paper contained a serious error. The current version is a complete reworking of the techniques and resultsDifferential GeometryAnalysis of PDEs53C17, 53A10, 49Q20, 35J70text