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

Abstract

We study the existence problem for tilted unduloids in H 2 × R. These are singly periodic annuli with constant mean curvature H > 1/2 in H 2 × 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 H 2 × 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 H 2 × R exist.