The Gauss Map of Minimal Surfaces in the Heisenberg Group

Abstract

Abstract. We study the Gauss map of minimal surfaces in the Heisenberg group Nil3 endowed with a left-invariant Riemannian metric. We prove that the Gauss map of a nowhere vertical minimal surface is harmonic into the hyperbolic plane H 2 . Conversely, any nowhere antiholomorphic harmonic map into H 2 is the Gauss map of a nowhere vertical minimal surface. Finally, we study the image of the Gauss map of complete nowhere vertical minimal surfaces.

Holomorphic quadratic differentials and the Bernstein problem in Heisenberg space

“…It is nonetheless very likely that our arguments can be modified in certain aspects to give an alternative proof of Theorem 1 without using H 2 × R at all. This possibility is given by the harmonicity of the Gauss map for minimal surfaces in Nil 3 [Dan2]. We have followed here the path across H 2 × R to give a natural continuation of the work [FeMi1], which is where the canonical examples were first constructed, and also because we believe that this more flexible perspective may give a clearer global picture of CMC surface theory in homogeneous 3-manifolds.…”

“…will always be assumed to be canonically oriented, meaning that its unit normal is upwards pointing. [Sa] (see also [FeMi1,Dan2] …”

“…Although the elements of this family are non-congruent in general (e.g., if the Gauss map has no symmetries), it happens that for minimal surfaces in Nil 3 invariant under a 1-parameter subgroup of isometries many of the elements of the 2-parameter family are congruent to each other. A detailed discussion on this topic can be found on page 1165 of [FeMi1] (see also [Dan2]). …”

“…Indeed, in [Dan2] Daniel gave an extension to the case of minimal surfaces in Nil 3 of the harmonicity theory for the hyperbolic Gauss map of H = 1/2 surfaces in H 2 ×R. In this Nil 3 setting, the harmonic Gauss map is simply the usual Gauss map of a surface with regular vertical projection (taking values on the upper hemisphere of the Lie algebra of Nil 3 ), composed with stereographic projection.…”

“…There are other explicit entire solutions to (1.1) foliated by Euclidean straight lines (see [Dan2]), which suggests that the class of entire minimal graphs in Nil 3 is quite large. The Bernstein problem in Nil 3 is a usual discussion topic among people working on CMC surfaces in homogeneous spaces, and appears explicitly as an open problem in several works such as [FMP,AbRo2,Dan2,IKOS,MMP]. Let us also point out that the Bernstein problem in the sub-Riemannian Heisenberg space H 1 has been intensely studied in the last few years (see [BSV, CHMY, GaPa, RiRo] and the references therein).…”