# On the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$

## TL;DR

This paper investigates the existence of a special class of constant mean curvature surfaces called tilted unduloids in hyperbolic space times a line, linking their existence to a uniqueness problem in Berger spheres.

## Contribution

It reduces the existence problem of tilted unduloids in \\mathbb{H}^2\\times\\mathbb{R} to a geometric uniqueness problem in Berger spheres using the Daniel correspondence.

## Key findings

- Existence of tilted unduloids depends on a uniqueness condition in Berger spheres.
- The problem is reduced to analyzing embedded minimal annuli bounded by linked geodesics.
- Provides a new approach to studying constant mean curvature surfaces in product spaces.

## Abstract

We study the existence problem for tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$. These are singly periodic annuli with constant mean curvature $H>1/2$ in $\mathbb{H}^2\times\mathbb{R}$, and the periodicity of these surfaces is with respect to a discrete group of translations along a geodesic that is neither vertical nor horizontal in the Riemannian product $\mathbb{H}^2\times\mathbb{R}$. Via the Daniel correspondence we are able to reduce this existence problem to a uniqueness problem in the Berger spheres: if a pair of linked horizontal geodesics bounds exactly two embedded minimal annuli (for a fixed orientation of the boundary curves) then tilted unduloids in $\mathbb{H}^2\times\mathbb{R}$ exist.