The Bernstein problem in Heisenberg groups
Abstract
In these notes, we collect the main and, to the best of our knowledge, most up-to-date achievements concerning the Bernstein problem in the Heisenberg group; that is, the problem of determining whether the only entire minimal graphs are hyperplanes. We analyze separately the problem for t-graphs and for intrinsic graphs: in the first case, the Bernstein Conjecture turns out to be false in any dimension, and a complete characterization of minimal graphs is available in H1 for the smooth case. A positive result is instead available for Lipschitz intrinsic graphs in H1; moreover, one can see that the conjecture is false in Hn with n at least 5, by adapting the Euclidean counterexample in high dimension; the problem is still open when n is 2, 3 or 4.
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