The Bernstein Problem 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.

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 results

References in corpus (1)

• Minimal surfaces in the Heisenberg group

Cited by in corpus (15)

• Minimal surfaces in the Heisenberg group
• Minimal surfaces in pseudohermitian geometry
• Area-stationary surfaces in the Heisenberg group H^1
• The Hessian of the distance from a surface in the Heisenberg group
• Variation of Perimeter Measure in sub-Riemannian geometry
• Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$
• Constant mean curvature surfaces in sub-Riemannian geometry
• H-minimal graphs of low regularity in the Heisenberg group
• Isoperimetric and Sobolev inequalities on hypersurfaces in sub-Riemannian Carnot groups
• Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model
• Properly embedded and immersed minimal surfaces in the Heisenberg group
• Straight ruled surfaces in the Heisenberg group
• Isoperimetric, Sobolev and Poincaré inequalities on hypersurfaces in sub-Riemannian Carnot groups
• Stable H-minimal hypersurfaces
• Generalized mean curvature flow in Carnot groups