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

Vincent Borrelli, Roland Denis, Francis Lazarus, Mélanie Theillière, Boris Thibert

Published 2023-03-22Version 1

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. Our construction generates iteratively a sequence of maps by adding at each step $k$ a layer of $N_{k}$ corrugations. In parallel, we introduce a formal corrugation process leading to a limit map $\Phi_{\infty}:\mathbb{H}^2\to \mathcal{L}(\mathbb{R}^2,\mathbb{R}^3)$ that we call the formal analogue of $df_{\infty}$. We show that $\Phi_{\infty}$ approximates $df_{\infty}$. In fact, when the sequence of corrugation numbers $(N_{k})_{k}$ grows to infinity the map $\Phi_{\infty}$ encodes the asymptotic behavior of $df_{\infty}$. Moreover, analysing the geometry of $\Phi_{\infty}$ appears much easier than studying $f_{\infty}$. In particular, we are able to exhibit a form of self-similarity of $\Phi_{\infty}$.