Half‐space theorem, embedded minimal annuli and minimal graphs in the Heisenberg group

By Benoît Daniel, L. Hauswirth. 2007

80 citations recorded by the source corpus.

We construct a one‐parameter family of properly embedded minimal annuli in the Heisenberg group Nil 3 endowed with a left‐invariant Riemannian metric. These annuli are not rotationally invariant. This family of annuli is used to prove a vertical half‐space theorem which is then applied to prove that each complete minimal graph in Nil 3 is entire. Also, it is shown that the sister surface of an entire minimal graph in Nil 3 is an entire constant mean curvature (CMC) ½ graph in ℍ2 × ℝ, and vice versa. This gives a classification of all entire CMC ½ graphs in ℍ2 × ℝ. Finally we construct properly embedded CMC ½ annuli in ℍ2 × ℝ.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/964b535cc15afc683297cca690d5dc7d13cb2029

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In Computational Theory and Mathematics

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Citations in the corpus

The Gauss Map of Minimal Surfaces in the Heisenberg Group (2006, methodology): “z and P. Mira solved the Bernstein problem in Nil3, i.e., they classiﬁed all entire minimal graphs in Nil3 (in terms of their Abresch-Rosenberg diﬀerential), THE GAUSS MAP OF MINIMAL SURFACES 17 • in [DH09], the author and L. Hauswirth proved that every complete nowhere vertical minimal surface in N”
Complete constant mean curvature surfaces and Bernstein type theorems in $M^2 x R$ (2008, background): “Entire constant mean curvature 1/2 graphs, in H2 × R and entire minimal graphs in Heisenberg space have been classified (they are sister surfaces, see [10], [11] and [15]).”