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 $df_\infty$ 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 $df_\infty$ is close to $\Phi_{\infty}$ up to a precision that depends on the sequence $N_*:= (N_{k})_k$. We then introduce the $pattern$ $maps$ $\boldsymbol{\nu}_{\infty}^\Phi$ and $\boldsymbol{\nu}_{\infty}$, of respectively $\Phi_{\infty}$ and $df_\infty$, that together with $df_0$ entirely describe the geometry of the Gauss maps associated to $\Phi_{\infty}$ and $df_\infty$. For well chosen sequences of corrugation numbers, we finally show an asymptotic convergence of $\boldsymbol{\nu}_{\infty}$ towards $\boldsymbol{\nu}_{\infty}^\Phi$ over circles of rational radii.