Minimal Graphs in Nil_3 : existence and non-existence results

TL;DR

This paper investigates the existence and non-existence of minimal graphs in the Heisenberg space Nil_3, providing solutions for certain boundary conditions and constructing specific minimal surfaces, while also highlighting open problems.

Contribution

It offers new existence and non-existence results for minimal graphs in Nil_3, including solutions over convex domains and the construction of Scherk type minimal surfaces.

Findings

Non-existence of minimal graphs over non-convex domains

Existence of solutions over convex domains with bounded, piecewise continuous boundary data

Construction of Scherk type minimal surfaces in Nil_3

Abstract

We study the minimal surface equation in the Heisenberg space, Nil_3. A geometric proof of non existence of minimal graphs over non convex, bounded and unbounded domains is achieved (our proof holds in the Euclidean space as well). We solve the Dirichlet problem for the minimal surface equation over bounded and unbounded convex domains, taking bounded, piecewise continuous boundary value. We are able to construct a Scherk type minimal surface and we use it as a barrier to construct non trivial minimal graphs over a wedge of angle between \pi/2 ,and \pi taking non negative continuous boundary data, having at least quadratic growth. In the case of an half- plane, we are also able to give solutions (with either linear or quadratic growth), provided some geometric hypothesis on the boundary data. Finally, some open problem arising from our work, are posed.