The Bernstein problem for intrinsic graphs in Heisenberg groups and calibrations

Abstract

In this paper we deal with some problems concerning minimal hypersurfaces in Carnot-Carathéodory (CC) structures. More precisely we will introduce a general calibration method in this setting and we will study the Bernstein problem for entire regular intrinsic minimal graphs in a meaningful and simpler class of CC spaces, i.e. the Heisenberg group $${\mathbb{H}^n}$$ . In particular we will positively answer to the Bernstein problem in the case n = 1 and we will provide counter examples when $${n\geq 5}$$ .