A Construction Method for Discrete Constant Negative Gaussian Curvature Surfaces

Abstract

This article is an application of the author’s paper (Kobayashi, Nonlinear d’Alembert formula for discrete pseudospherical surfaces, 2015, [9]) about a construction method for discrete constant negative Gaussian curvature surfaces, the nonlinear d’Alembert formula. The heart of this formula is the Birkhoff decomposition, and we give a simple algorithm for the Birkhoff decomposition in Lemma 3.1. As an application, we draw figures of discrete constant negative Gaussian curvature surfaces given by this method (Figs. 1 and 2).

Appendix

In this appendix we give a definition of the loop group of $\mathrm{SU}_2$ and its subgroups $\varLambda ^{\pm } \mathrm{SU}_2$. Moreover theorem of the Birkhoff decomposition will be stated.

It is easy to see that F defined in (3) together with the condition $F|_{(x_*, y_*)} = {\text {Id}}$ is an element in the twisted $\mathrm{SU}_2$ -loop group :

$$\begin{aligned} \varLambda \mathrm{SU}_2:= \left\{ g : \mathbb R^{\times } \cup S^1 \rightarrow \mathrm{SL}_2 \mathbb C\;\Big |\; \begin{array}{l} g\ \text {is smooth}, g(\lambda ) = \left( \overline{g(\bar{\lambda })}^{-1} \right) ^{T} \\ \;\text {and}\ \sigma g(\lambda ) = g(-\lambda ) \end{array}\right\} , \end{aligned}$$

(20)

where $\mathbb R^{\times } =\mathbb R{\setminus }\{0\}$, $\sigma X = {\text {Ad}}(\sigma _3) X = \sigma _3 X \sigma _3^{-1}, (X \in \mathrm{SL}_2 \mathbb C)$ is an involution on $\mathrm{SL}_2 \mathbb C$. In order to make the above group a Banach Lie group, we restrict the occurring matrix coefficients to the Wiener algebra $\mathscr {A} =\{ f(\lambda )= \sum _{n \in \mathbb Z} f_n \lambda ^n{:}\,S^1 \rightarrow \mathbb C \;|\; \sum _{n \in \mathbb Z} |f_n| < \infty \}$, where we denote the Fourier expansion of f on $S^1$ by $f(\lambda )=\sum _{n \in \mathbb Z} f_n \lambda ^n$. Then the Wiener algebra is a Banach algebra relative to the norm $\Vert f\Vert = \sum |f_n|$ and the loop group $\varLambda \mathrm{SU}_2$ is a Banach Lie group, [6].

Let $\mathbb D^{+}$ and $\mathbb D^{-}$ be the interior of the unit disk in the complex plane and the union of the exterior of the unit disk in the complex plane and infinity, respectively. We first define two subgroups of $\varLambda \mathrm{SU}_2$:

$$\begin{aligned} \varLambda ^+ \mathrm{SU}_2=\left\{ g \in \varLambda \mathrm{SU}_2\;|\;\text{ g } \text{ can } \text{ be } \text{ analytically } \text{ extended } \text{ to } \mathbb D^+ \right\} , \end{aligned}$$

(21)

$$\begin{aligned} \varLambda ^- \mathrm{SU}_2=\left\{ g \in \varLambda \mathrm{SU}_2\;|\;\text{ g } \text{ can } \text{ analytically } \text{ be } \text{ extended } \text{ to } \mathbb D^{-} \right\} . \end{aligned}$$

(22)

Then $\varLambda ^+_{*} \mathrm{SU}_2$ and $\varLambda ^-_{*} \mathrm{SU}_2$ denote subgroups of $\varLambda ^+ \mathrm{SU}_2$ and $\varLambda ^- \mathrm{SU}_2$ normalized at $\lambda =0$ and $\lambda = \infty$, respectively:

$$\begin{aligned} \varLambda ^+_{*} \mathrm{SU}_2=\left\{ g \in \varLambda ^+ \mathrm{SU}_2\;|\; g(\lambda =0) = {\text {Id}}\right\} ,\\ \varLambda ^-_{*} \mathrm{SU}_2=\left\{ g \in \varLambda ^- \mathrm{SU}_2\;|\; g(\lambda =\infty ) = {\text {Id}}\right\} . \end{aligned}$$

The following decomposition theorem is fundamental.

Theorem 3.3

(Birkhoff decomposition, [3, 6]) The multiplication maps

$$\begin{aligned} \varLambda ^+_{*} \mathrm{SU}_2\times \varLambda ^- \mathrm{SU}_2\rightarrow \varLambda \mathrm{SU}_2\;\;\text{ and }\;\; \varLambda ^-_{*} \mathrm{SU}_2\times \varLambda ^+ \mathrm{SU}_2\rightarrow \varLambda \mathrm{SU}_2\end{aligned}$$

are diffeomorphisms onto $\varLambda \mathrm{SU}_2$, respectively.