The Hyperbolic Plane in ${\mathbb{E}}^3$

Abstract

We build an explicit $C^1$ isometric embedding $f_{\infty }:{\mathbb{H}}^2\to {\mathbb{E}}^3$ of the hyperbolic plane whose image is relatively compact. Its limit set is a closed curve of Hausdorff dimension 1. Given an initial embedding $f_0$, our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_k$ corrugations. To understand the behavior of $d{f}_{\infty }$ we introduce a $formal$ $corrugation$ $process$ leading to a $formal$ $analogue$ ${\Phi }_{\infty }:{\mathbb{H}}^2\to \mathcal{L}({\mathbb{R}}^2,{\mathbb{R}}^3)$. We show a self-similarity structure for ${\Phi }_{\infty }$. We next prove that $d{f}_{\infty }$ is close to ${\Phi }_{\infty }$ up to a precision that depends on the sequence $N_{\ast }:=(N_k{)}_k$. We then introduce the $pattern$ $maps$ ${\mathbf{\nu }}_{\infty }^{\Phi }$ and ${\mathbf{\nu }}_{\infty }$, of respectively ${\Phi }_{\infty }$ and $d{f}_{\infty }$, that together with $d{f}_0$ entirely describe the geometry of the Gauss maps associated to ${\Phi }_{\infty }$ and $d{f}_{\infty }$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of ${\mathbf{\nu }}_{\infty }$ towards ${\mathbf{\nu }}_{\infty }^{\Phi }$ over circles of rational radii.

51 pages, 8 figures