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.
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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
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