An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$

Abstract

In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in H 2 × R. As a consequence, we prove that there is no properly immersed minimal surface whose asymptotic boundary ∞ is a Jordan curve homologous to zero in ∂ ∞ H 2 × R such that ∞ is contained in a slab between two horizontal circles of ∂ ∞ H 2 × R with width equal to π. We construct vertical minimal graphs in H 2 ×R over certain unbounded admissible domains taking certain prescribed finite boundary data and certain prescribed asymptotic boundary data. Our admissible unbounded domains in H 2 × {0} are non necessarily convex and non necessarily bounded by convex arcs; each component of its boundary is properly embedded with zero, one or two points on its asymptotic boundary, satisfying a further geometric condition.

New complete embedded minimal surfaces in $${{\mathbb {H} ^2\times \mathbb {R}}}$$

“…We refer to the existence theorem of minimal surfaces. Theorem 2.1 (Corollary 4.1 of [15]) Let ⊂ H 2 ×{0} be a domain and let g : ∂ ∪∂ ∞ → R be a bounded function everywhere continuous except perhaps at a finite set S ⊂ ∂ ∪∂ ∞ . Assume that the finite boundary ∂ is convex.…”

A halfspace theorem for mean curvature H=12 surfaces in H2×R

“…A halfspace theorem for minimal surfaces in H 2 × R is false, in fact there are many vertically bounded complete minimal surfaces in H 2 × R [11]. On the contrary, we are able to prove the following result for H = 1 2 surfaces in H 2 × R. The result in [5] is different in nature from our result because in [5], the "halfspace" is one side of a horocylinder, while for us, the "halfspace" is the mean convex side of a rotational simply connected surface.…”

Uniform a priori estimates for a class of horizontal minimal equations

“…A simple verification shows that if then the first eigenvalue of the horizontal minimal operator goes to zero. On the other hand, it is amazing that the vertical minimal equation is strictly elliptic for all values of the independent variables [, Equation 4], [, Equation 6].…”

“…Notice that there is a non‐existence result that follows from the asymptotic principle proved by E. Toubiana and the author in [, Theorem 2.1]. Namely, there is no horizontal minimal graph given by a function on a bounded smooth strictly convex domain Ω, taking zero boundary data on Furthermore, there is no horizontal minimal graph in , over a bounded Jordan domain Ω strictly contained in an horizontal slab of height π, given by a function taking zero asymptotic boundary data on [, Corollary 2.1].…”

“…The first main results about minimal vertical graphs were derived by B. Nelli and H. Rosenberg . Since then, a significant progress in the theory has been achieved, see, for instance, , , , . When the ambient space is the notion of vertical mean curvature equation in n independent variables has also been established and developed , , , .…”