Minimal graphs in $$Nil_3:$$ N i l 3 : existence and non-existence results

Abstract

We study the minimal surface equation in the Heisenberg space, N il3. 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 θ ∈ [ π 2 , π[ taking non negative continuous boundary data, having at least quadratic growth. In the case of an halfplane, 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.

Height and Area Estimates for Constant Mean Curvature Graphs in $$\mathbb {E}(\kappa ,\tau )$$ E ( κ , τ ) -Spaces

“…Fernández and Mira [14] showed that there exists a vast family of entire minimal graphs in Nil 3 (τ ), namely, they can associate to each holomorphic quadratic differential Q on C or D = {z ∈ C : |z| < 1} a 2-parameter family of entire minimal graphs with Abresch-Rosenberg differential Q. The only restriction is Q = 0 if the domain is C. [26] proved that, for any wedge S with vertex at the origin and angle θ ∈] π 2 , π[, there exists a non-zero minimal graph over S, with zero boundary value. Here the proof is based on classical PDE's theory joint with a suitable construction of barriers.…”

“…This result is sharp, as half of a catenoid or planes of the form u(x, y) = ax + by show. Other non-trivial examples of graphs over a sector with angle between π 2 and π, with either linear or at least quadratic height growth, are given in [3] and [26] (see also Section 3).…”

“…In particular, if the height of an entire minimal graph Σ ⊂ Nil 3 (τ ) grows at most quadratically with respect to the distance to the origin in the base R 2 , then Σ has exactly cubic extrinsic area growth (Corollary 3). Height is always measured with respect to the usual zero section in Nil 3 (τ ), so this situation applies to many known explicit examples of entire minimal graphs (e.g., see [3,9,16,26]). In this sense, the last part of the paper deals with height estimates for minimal graphs in Nil 3 (τ ).…”

Uniform a priori estimates for a class of horizontal minimal equations

“…At last, the general a priori gradient estimates carry out by H. Rosenberg, E. Toubiana and R. Souam [, Theorem 3.6] has been applied to solve Dirichlet problems in homogeneous spaces .…”

Dual quadratic differentials and entire minimal graphs in Heisenberg space

“…Although this was already known, see [2,8,16], our approach also shows that invariant examples are related by hyperbolic deformations. Some other invariant examples will be discussed throughout the paper, though very few examples of explicit entire minimal graphs are known so far (e.g., the examples constructed by Cartier [5] or Nelli, Sa Earp and Toubiana [30] fail to be explicit).…”