Generic non-uniqueness of complete $H$-surfaces embedded in $\mathbb{H}^3$

TL;DR

This paper demonstrates that for most simple closed curves at infinity in hyperbolic space, there are multiple complete constant mean curvature surfaces bounded by these curves, highlighting non-uniqueness in such geometric configurations.

Contribution

It establishes the generic non-uniqueness of complete embedded H-surfaces in hyperbolic space for |H|<1, contrasting with the uniqueness for smoother boundary curves.

Findings

Most simple closed curves bound multiple H-surfaces.

Non-uniqueness is generic in the supremum metric topology.

This non-uniqueness does not hold for C^1 curves.

Abstract

We prove that, given $|H|<1$, a generic simple closed curve embedded in the asymptotic boundary of $\mathbb{H}^3$ (with respect to the supremum metric) bounds more than one complete surface embedded in $\mathbb{H}^3$ which has constant mean curvature $H$. We remark that this is not true for the space of simple closed $\text{C}^1$-curves.