Instability of graphical strips and a positive answer to the Bernstein   problem in the Heisenberg group
  Let S be a C^2 H-minimal noncharacteristic hypersurface in the first Heisenberg group. We show that if S contains a graphical strip, then it is not a stable minimal surface. Moreover, we show that if S is a C^2 H-minimal noncharacteristic entire graph which is not itself a vertical plane, then S contains a graphical strip. Thus, as a corollary, we obtain an analogue of the Bernstein theorem: the only stable C^2 H-minimal noncharacteristic entire graphs are the vertical planes. 
D. Danielli, N. Garofalo, D. M. Nhieu and S. D. Pauls