# Families of Minimal Surfaces in $$\mathbb {H}^2 \times \mathbb {R}$$ Foliated by Arcs and Their Jacobi Fields

@article{Ferrer2017FamiliesOM,
title={Families of Minimal Surfaces in \$\$\mathbb \{H\}^2 \times \mathbb \{R\}\$\$ Foliated by Arcs and Their Jacobi Fields},
author={Leonor Ferrer and Francisco Jos'e Plaza Mart'in and Rafe Mazzeo and Magdalena Rodr'iguez},
journal={arXiv: Differential Geometry},
year={2017},
pages={67-88}
}
This note provides some new perspectives and calculations regarding an interesting known family of minimal surfaces in $$\mathbb {H}^2 \times \mathbb {R}$$. The surfaces in this family are the catenoids, parabolic catenoids and tall rectangles. Each is foliated by either circles, horocycles or circular arcs in horizontal copies of $$\mathbb {H}^2$$. All of these surfaces are well-known, but the emphasis here is on their unifying features and the fact that they lie in a single continuous family… Expand

#### References

SHOWING 1-10 OF 13 REFERENCES
Properly embedded minimal annuli in $$\mathbb {H}^2 \times \mathbb {R}$$H2×R
• Mathematics
• Mathematische Annalen
• 2019
In this paper we study the moduli space of properly Alexandrov-embedded, minimal annuli in $$\mathbb {H}^2 \times \mathbb {R}$$H2×R with horizontal ends. We say that the ends are horizontal when theyExpand
An asymptotic theorem for minimal surfaces and existence results for minimal graphs in $${\mathbb H^2 \times \mathbb R}$$
• Mathematics
• 2008
In this paper we prove a general and sharp Asymptotic Theorem for minimal surfaces in $${\mathbb H^2 \times \mathbb R}$$. As a consequence, we prove that there is no properly immersed minimal surfaceExpand
“Minimal Surfaces in ℍ2 × ℝ”
• Mathematics
• 2002
In ℍ2 × ℝ” one has catenoids, helicoids and Scherk-type surfaces. A Jenkins-Serrin type theorem holds here. Moreover there exist complete minimal graphs in ℍ2 with arbitrary continuous asymptoticExpand
Isometric immersions into ⁿ×ℝ and ℍⁿ×ℝ and applications to minimal surfaces
We give a necessary and sufficient condition for an n-dimensional Riemannian manifold to be isometrically immersed in S n x ℝ or ℍ n × ℝ in terms of its first and second fundamental forms and of theExpand
Minimal surfaces of Riemann type in three-dimensional product manifolds
We construct and classify minimal surfaces foliated by horizontal curves of constant curvature in H2×R, R2×R and S2×R. The main tool is the existence of a Shiffman Jacobi field; such fieldsExpand
On the asymptotic behavior of minimal surfaces in H²×R
• Mathematics
• 2015
We consider the asymptotic behavior of properly embedded minimal surfaces in the product of the hyperbolic plane with the line, taking into account the fact that there is more than one naturalExpand
The Dirichlet problem for the minimal surface equation -with possible infinite boundary data- over domains in a Riemannian surface
• Mathematics
• 2011
In this paper, we study existence and uniqueness of solutions to Jenkins-Serrin type problems on domains in a Riemannian surface. In the case of unbounded domains, the study is focused on theExpand
An asymptotic theorem for minimal surfaces and existence results for minimal graphs in H2
• 2008
Minimal surfaces in H2×R
• Bull. Braz. Math. Soc. (N.S.)
• 2002