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One of the main objectives of this paper is to unravel it new interesting phenomenon of the sub-Riemannian Bernstein problem with respect to its Euclidean ancestor, with the purpose of also indicating a possible line of attack toward its solution. We show that the global intrinsic graphs (1.2) are unstable critical points of the horizontal perimeter. As a consequence of this fact, the study of the stability acquires a central position in the problem itself

A notable family of entire intrinsic minimal graphs in the Heisenberg group which are not perimeter minimizing