Instability of graphical strips and a positive answer to the Bernstein problem in the Heisenberg group H1

By D. Danielli, Nicola Garofalo, D. Nhieu, Scott D. Pauls. 2006

80 citations recorded by the source corpus.

In the first Heisenberg group H 1 with its sub-Riemannian structure generated by the horizontal subbundle, we single out a class of C 2 non-characteristic entire intrinsic graphs which we call strict graphical strips. We prove that such strict graphical strips have vanishing horizontal mean curvature (i.e., they are H-minimal) and are unstable (i.e., there exist compactly supported deformations for which the second variation of the horizontal perimeter is strictly negative). We then show that, modulo left-translations and rotations about the center of the group, every C 2 entire Hminimal graph with empty characteristic locus and which is not a vertical plane contains a strict graphical strip. Combining these results we prove the conjecture that in H 1 the only stable C 2 Hminimal entire graphs, with empty characteristic locus, are the vertical planes.

Subfields: Applied Mathematics, Genetics

View Paper: https://www.semanticscholar.org/paper/d0db5b8f69f2e8aee0b38e7579ba80f5544bda86

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Citations in the corpus

Intrinsic regular graphs in Heisenberg groups vs. weak solutions of non-linear first-order PDEs (2009, background): “The latter has been in the framework of minimal surfaces in Hn (see [24],[10], [3],[ 11 ], [6] and [7]).”
Sub-Riemannian calculus on hypersurfaces in Carnot groups (2006, background): “On the other hand, it has also been shown in [28] that every entire H -minimal graph in H1, with empty characteristic locus, and which is not itself a vertical plane ax + by = γ , after possibly a left-translation and a rotation about the t-axis, contains a graphical strip of the type x = yG(t), wit”
Regularity of non-characteristic minimal graphs in the Heisenberg group $H^1$ (2008, background): “Riemannian regularizations of sub-Laplacians (see the statement in Theorem 3.3). For other aspects of minimal surfaces in the Heisenberg group, including classiﬁcation and Bernstein-type results, see [8, 27, 2, 22, 21, 37, 36, 35]. These works also contain more comprehensive lists of references. Ack”