Uniqueness of H-surfaces in $${\mathbb{H}}^2 \times \mathbb{R},{{\vert H\vert \leq 1/2}}$$ , with boundary one or two parallel horizontal circles

Abstract

We prove that a H -surface M in H 2 × R, |H | 1/2, inherits the symmetries of its boundary ∂ M, when ∂ M is either a horizontal curve with curvature greater than one or two parallel horizontal curves with curvature greater than one, whose distance is greater or equal to π. Furthermore we prove that the asymptotic boundary of a surface with mean curvature bounded away from zero consists of parts of straight lines, provided it is sufficiently regular.

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

“…We observe that these examples are a particular case of a general result found in [3]. The proof of Lemma 5.1 follows from Proposition 5.1 of [14] and from [18]. For later use we call M + ρ (resp.…”

“…Let C ⊂ H 2 × R be a complete catenoid whose a component of the asymptotic boundary stays at height T 1 and the other component at height T 2 such that T 1 < a < b < T 2 (such a catenoid exists since 0 < b −a < π), note that T 2 − T 1 < π, the reader can see the geometric behaviour of the catenoids in Lemma 5.1 or [14]. By continuity, we can choose T 1 and T 2 such that M 0 is entirely contained in the open slab H 2 × (T 1 , T 2 ).…”

Non-properly embedded H-planes in $${\mathbb H}^2\times {\mathbb R}$$ H 2 × R

“…We begin this section by recalling several results in [8,9]. Given H ∈ (0, 1 2 ) and d ∈ [−2H, ∞), let…”

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

“…Sa Earp and E. Toubiana find explicit integral formulas for rotational surfaces of constant mean curvature H ∈ (0, 1 2 ] in [10]. A careful description of the geometry of these surfaces is contained in Lemma 5.2 and Proposition 5.2 in the Appendix of [8].…”