Abstract
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. We also initiate a study of the Jacobi operator on the parabolic catenoid, and compute the Jacobi fields associated to deformations to either of the two other types of surfaces in this family.
L. Ferrer, F. Martín and M. M. Rodríguez—are partially supported by MINECO/FEDER grant MTM2014-52368-P and MTM2017-89677-P.
F. Martin—is also partially supported by the Leverhulme Trust grant IN-2016-019.
R. Mazzeo—is supported by the NSF grant DMS-1608223.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
References
Daniel, B.: Isometric immersions into \({\mathbb{S}}\mathit{^{n}\times {\mathbb{R}}}\) and \({\mathbb{H}}\mathit{^{n}\times {\mathbb{R}}}\) and applications to minimal surfaces. Trans. A.M.S. 361, 6255–6282 (2009)
Ferrer, L., Martín, F., Mazzeo, R., Rodríguez, M.: Properly embedded minimal annuli in \({\mathbb{H}}\mathit{^{2} \times {\mathbb{R}}}\). Math. Ann. 375, 541–594 (2019)
Hauswirth, L.: Minimal surfaces of Riemann type in three-dimensional product manifolds. Pacific J. Math. 224, 91–117 (2006)
Kloeckner, B., Mazzeo, R.: On the asymptotic behavior of minimal surfaces in \(\mathbb{H}^{2}\times \mathbb{R}\). Indiana Math J. 66(2), 631–658 (2017)
Mazet, L., Rodríguez, M., Rosenberg, H.: The Dirichlet problem for the minimal surface equation -with possible infinite boundary data- over domains in a Riemannian surface. Proc. Lond. Math. Soc. 102(3), 985–023 (2011). https://doi.org/10.1112/plms/pdq032. arXiv:0806.0498
Nelli, B., Rosenberg, H.: Minimal surfaces in \({\mathbb{H}}\mathit{^{2} \times {\mathbb{R}}}\). Bull. Braz. Math. Soc. (N.S.) 33(2), 263–292 (2002)
Sa Earp, R., Toubiana, E.: An asymptotic theorem for minimal surfaces and existence results for minimal graphs in \({\mathbb{H}}\mathit{^2 \times {\mathbb{R}}}\). Math. Ann. 342 309–331 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Ferrer, L., Martín, F., Mazzeo, R., Rodríguez, M. (2021). Families of Minimal Surfaces in \(\mathbb {H}^2 \times \mathbb {R}\) Foliated by Arcs and Their Jacobi Fields. In: Hoffmann, T., Kilian, M., Leschke, K., Martin, F. (eds) Minimal Surfaces: Integrable Systems and Visualisation. m:iv m:iv m:iv m:iv 2017 2018 2018 2019. Springer Proceedings in Mathematics & Statistics, vol 349. Springer, Cham. https://doi.org/10.1007/978-3-030-68541-6_5
Download citation
DOI: https://doi.org/10.1007/978-3-030-68541-6_5
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-68540-9
Online ISBN: 978-3-030-68541-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)