# Families of minimal surfaces in $\mathbb{H}^2 \times \mathbb{R}$ foliated by arcs and their Jacobi fields

## TL;DR

This paper explores a unified family of minimal surfaces in hyperbolic space, analyzing their geometric properties and Jacobi fields, with a focus on the parabolic catenoid and related deformations.

## Contribution

It unifies known minimal surfaces in ^2 imes \u211d into a continuous family and studies their Jacobi operators and associated deformation fields.

## Key findings

- Identification of a continuous family of minimal surfaces in ^2 imes 
- Calculation of Jacobi fields for deformations within this family
- Insights into the geometric and stability properties of these surfaces

## 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.