The Bernstein Problem in Heisenberg Groups

In these notes we summarize what is known about the Bernstein's problem in the Heisenberg group: that is, the problem of determining whether every entire area-minimizing graph of codimension 1 in H-n is a hyperplane. We analyze separately the problem for t-graphs and for horizontal graphs. For what concerns t-graphs, the conjecture has long been known to be false in H-1, even in the case of C-2 regularity (see [22]); we show that it is also false in higher dimensions. A positive result for Lipschitz intrinsic graphs in H-1 was achieved in [29], while counter-examples are available in H-n with n >= 5; the problem is still open for intrinsic graphs in H-n with n = 2, 3, 4.