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.

The Bernstein Problem in Heisenberg Groups / Serra Cassano, F; Vedovato, M. - In: LE MATEMATICHE. - ISSN 0373-3505. - 75:1(2020), pp. 377-403. [10.4418/2020.75.1.17]

The Bernstein Problem in Heisenberg Groups

Serra Cassano, F;Vedovato, M
2020

Abstract

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.
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Serra Cassano, F; Vedovato, M
The Bernstein Problem in Heisenberg Groups / Serra Cassano, F; Vedovato, M. - In: LE MATEMATICHE. - ISSN 0373-3505. - 75:1(2020), pp. 377-403. [10.4418/2020.75.1.17]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11572/284296
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