Mobius Transformation is Continuous From C Infinity to C Infinity

Mobius Transformation

Möbius transformations provide a natural family of intertwining operators for ρ1 coming from inner automorphisms of SL(2, ℝ) (will be used later).

From: North-Holland Mathematics Studies , 2004

Topics in Multivariate Approximation and Interpolation

Georg Zimmermann , in Studies in Computational Mathematics, 2006

2.1 The group and its Haar measure

Möbius transformations are complex functions of the form $z\mapsto \frac{\mathit{az}+b}{\mathit{cz}+d}$ , where a, b, c, d  ∈ ℂ with a d − b c ≠ 0. They are the biholomorphic maps $\overline{ℂ}\to \overline{ℂ}$ and form a group under concatenation. A peculiarity of this group is the fact that the mapping

$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\mapsto \left(z\mapsto \frac{\mathit{az}+b}{\mathit{cz}+d}\right)$

is a group homomorphism from GL ₂(ℂ). Since $\frac{\mathit{az}+b}{\mathit{cz}+d}\equiv \frac{a^{\prime }z+b^{\prime }}{c^{\prime }z+d^{\prime }}$ if and only if $\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)=\lambda \left(\begin{array}{cc}\hfill a^{\prime }\hfill & \hfill b^{\prime }\hfill \\ \hfill c^{\prime }\hfill & \hfill d^{\prime }\hfill \end{array}\right)$ for some λ  ∈ ℂ*, the group of Möbius transformations is isomorphic to the quotient of GL ₂(ℚ) or SL ₂(ℚ) by their respective center, usually referred to as PGL ₂(ℂ) or PSL ₂(ℂ).

To determine the subgroup G which maps the unit circle to itself and preserves the orientation, note that since |z| = 1 is equivalent to $z=1/\overline{z}$ , an element g  ∈ G has to satisfy

$g\left(z\right)=\frac{\mathit{az}+b}{\mathit{cz}+d}=1/\overline{g\left(z\right)}=\frac{\overline{c}\overline{z}+\overline{d}}{\overline{a}\overline{z}+\overline{b}}=\frac{\overline{c}+\overline{d}z}{\overline{a}+\overline{b}z}\mathrm{for}\mathrm{all}z\mathrm{with}z=1/\overline{z}\text{.}$

Since a holomorphic function is determined by its values on the set {|z| = 1}, we must have

$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)=\lambda \left(\begin{array}{cc}\hfill \overline{d}\hfill & \hfill \overline{c}\hfill \\ \hfill \overline{b}\hfill & \hfill \overline{a}\hfill \end{array}\right)$

which implies $\lambda \overline{\lambda }=1$ . Fixing a value for $\sqrt{\lambda }$ , we define $\alpha =\alpha /\sqrt{\lambda }$ and $\beta =b/\sqrt{\lambda }$ which yields $c/\sqrt{\lambda }=\sqrt{\lambda }\overline{b}=\overline{\beta }$ and $d/\sqrt{\lambda }=\sqrt{\lambda }\overline{a}=\overline{a}$ , i.e., we have

$g\left(z\right)=\frac{\mathit{\alpha z}+\beta }{\overline{\beta }z+\overline{\alpha }}\mathrm{for}\mathrm{some}\alpha ,\beta \in ℂ\mathrm{with}{\left|\alpha \right|}^2-{\left|\beta \right|}^2\ne 0\text{.}$

Since we want g to preserve the orientation of the unit circle, it has to map the unit disk onto itself, so $\left|g\left(0\right)\right|=\left|\beta /\overline{\alpha }\right|<1,\mathrm{i}.\mathrm{e}.,\left|\alpha \right|>\left|\beta \right|$ . Thus we may impose the normalization |α|² − |β|² = 1, which determines the pair (α, β) up to a factor of ±   1.

In other words, G is isomorphic to the quotient group of

$S{U}_{1,1}=\left\{\left(\frac{\alpha }{\beta }\frac{\beta }{\alpha }\right):{\left|\alpha \right|}^2-{\left|\beta \right|}^2=1\right\}$

by its center {± I ₂}, usually referred to as PSU _1,1, and we can use SU _1,1 as a double cover for G.

In particular, Haar measure on G is equivalent to Haar measure on SU _1,1, and to integrate over G we may integrate over SU _1,1 (the extra factor of two simply means choosing a different normalization of the Haar measure). To be able to integrate over SU _1,1, we use the map

$\begin{array}{cc}\hfill {ℝ}^{+}\times \mathsf{T}\times \mathsf{T}\to S{U}_{1,1}\hfill & \hfill \left(s,\varphi ,\theta \right)\mapsto \left(\begin{array}{cc}\hfill e^{2\mathit{\pi i\varphi }}cosh\left(s\right)\hfill & \hfill e^{2\mathit{\pi i\theta }}sinh\left(s\right)\hfill \\ \hfill e^{-2\mathit{\pi i\theta }}sinh\left(s\right)\hfill & \hfill e^{-2\mathit{\pi i\varphi }}cosh\left(s\right)\hfill \end{array}\right)\hfill \end{array}\text{.}$

This map covers SU _1,1, except for the set $\left\{\left(\begin{array}{cc}\hfill e^{2\mathit{\pi i\varphi }}\hfill & \hfill 0\hfill \\ \hfill 0\hfill & \hfill e^{-2\mathit{\pi i\varphi }}\hfill \end{array}\right):\varphi \in \mathsf{T}\right\}$ which is a submanifold of codimension 2 and as such a set of measure zero. We leave it as an exercise to the reader to show that both the left and the right Haar measure for SU _1,1 on this map are given by

$\mathit{dm}\left(s,\varphi ,\theta \right)=sinh2s\mathit{ds}\mathit{d\varphi }\mathit{d\theta }\text{,}$

so in particular, SU _1,1 and thus G is unimodular. We will frequently use the substitution

$\left(u,v\right)=\left(e^{2\mathit{\pi i\varphi }},e^{2\mathit{\pi i\theta }}\right)\mathrm{with}\mathit{d\varphi }\mathit{d\theta }=\frac{\mathit{du}}{2\mathit{\pi iu}}\frac{\mathit{dv}}{2\mathit{\pi iv}}\text{.}$

To simplify notation, we shall write (α, β) for $\left(\frac{\alpha }{\beta }\frac{\beta }{\alpha }\right)$ , and use the abbreviation

$m_{\left(\alpha ,\beta \right)}\left(z\right):=\frac{\mathit{\alpha z}+\beta }{\overline{\beta }z+\overline{\alpha }}\mathrm{for}\mathrm{any}\alpha ,\beta \in ℂ\mathrm{with}{\left|\alpha \right|}^2-{\left|\beta \right|}^2\ne 0\text{.}$

The following isomorphism should at least be mentioned here. We have SL ₂(ℝ)   ≅ SU _1,1 via

$\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\mapsto \frac{1}{2}\left(\begin{array}{cc}\hfill 1\hfill & \hfill -i\hfill \\ \hfill -i\hfill & \hfill 1\hfill \end{array}\right)\left(\begin{array}{cc}\hfill a\hfill & \hfill b\hfill \\ \hfill c\hfill & \hfill d\hfill \end{array}\right)\left(\begin{array}{cc}\hfill 1\hfill & \hfill i\hfill \\ \hfill i\hfill & \hfill 1\hfill \end{array}\right)=\frac{1}{2}\left(\begin{array}{cc}\hfill \left(a+d\right)+i\left(b-c\right)\hfill & \hfill \left(b+c\right)+i\left(a-d\right)\hfill \\ \hfill \left(b+c\right)+i\left(a-d\right)\hfill & \hfill \left(a+d\right)+i\left(b-c\right)\hfill \end{array}\right)\text{,}$

i.e., they are conjugate subgroups of SL ₂(ℂ). This carries over to PSL ₂(ℝ)   ≅ PSU _1,1, which geometrically means the following. PSL ₂(ℂ) represents the subgroup of Möbius transformations mapping the real line to itself, preserving orientation. Since the Möbius transformation

$z\mapsto \frac{z+i}{\mathit{iz}+1}$

maps the unit circle to the real line and the unit disk to the upper half plane, it intertwines the two groups.

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Introduction to complex dynamics

Á. Alberto Magreñán , Ioannis K. Argyros , in A Contemporary Study of Iterative Methods, 2018

21.3 Topological conjugations

This section will focus attention to the topological conjugations which will help us in the study of the dynamics of iterative methods. Before introducing this concept we need the definition of a Möbius transformation.

Definition 21.3.1

A Möbius transformation of parameters $a,b,c,d\in \mathbb{C}$ is a function Γ with the following form

$\mathrm{\Gamma }(z)=\frac{az+b}{cz+d},\forall z\in \hat{\mathbb{C}}$

where $a,b,c,d$ are such that $ad-bc\ne 0$ .

Definition 21.3.2

Let $R_1,R_2:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ be two rational functions. We say that they are conjugated if and only if there exists a Möbius transformation $\phi :\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ such that $\phi \circ R_1\circ {\phi }^{-1}(z)=R_2(z)$ for each z.

If $R_1$ and $R_2$ are conjugated by M, then it is clear that

$M(\mathcal{J}(R_1))=\mathcal{J}(R_2)$

and

$M(\mathcal{F}(R_1))=\mathcal{F}(R_2).$

The properties of the topological conjugations have been briefly studied by many authors, e.g., Beardon [8] and Fagella et al. [12]. In the following result a short resumé of these properties is shown (see Fig. 21.3).

Figure 21.3. Diagram of a topological conjugation.

Theorem 21.3.3

Let $R_1,R_2:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ be two rational functions and let $\phi :\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ be a topological conjugation between $R_1(z)$ and $R_2(z)$ . Then the following assertions hold:

(i)

${\phi }^{-1}:\hat{\mathbb{C}}\to \hat{\mathbb{C}}$ is also a topological conjugation between $R_1(z)$ and $R_2(z)$ .

(ii)

$\phi \circ R_1^n(z)=R_2^n\circ \phi (z)$ , for each $n\in \mathbb{N}$ .

(iii)

p is a periodic point of $R_1(z)$ if and only if $\phi (p)$ is a periodic point of $R_2(z)$ . Moreover, both periodic points have the same period.

(iv)

If p is a periodic point of $R_1(z)$ and ${\phi }^{\prime }(z)\ne 0$ for each $z\in \mathcal{O}(p)$ , then p and $\phi (p)$ have the same behavior.

(v)

If p is a periodic point of $R_1(z)$ with basin of attraction $\mathcal{A}(p)$ , then the basin of attraction of $\phi (p)$ is $\phi (\mathcal{A}(p))$ .

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Geometric Function Theory

Samuel L. Krushkal , in Handbook of Complex Analysis, 2005

2.7.3

Let us look at some applications of the obtained results.

First consider the ordered quadruples a = (a ₁, a ₂, a ₃, a ₄) of distinct points on ${\displaystyle \widehat{\mathbb{C}}}$ . Their cross-ratios

(2.28) $\alpha =\frac{a_3-a_1}{a_2-a_1}:\frac{a_3-a_4}{a_2-a_4}\in \mathbb{C}\backslash \{0,1\}=:{\mathbb{C}}^{*}$

are invariant under Möbius transformations. Let $\mathbb{C}(a)={\displaystyle \widehat{\mathbb{C}}}\backslash \{a_1,a_2,a_3,a_4\}$ . The following Teichmüller theorem [Te1] has various applications in the theory of quasiconformal maps.

Theorem 2.5. There is a K-quasiconformal automorphism of ${\displaystyle \widehat{\mathbb{C}}}$ moving the ordered quadruple (a ₁, a ₂, a ₃, a ₄) into the ordered quadruple $(a_1^{\prime },a_2^{\prime },a_3^{\prime },a_4^{\prime })$ if and only if their cross-ratios α and α′ satisfy ${\rho }_{{\mathbb{C}}^{*}}(\alpha ,{\alpha }^{\prime })⩽\frac{1}{2}\log K$ , where ${\rho }_{\mathbb{C}*}(\cdot ,\cdot )$ is the hyperbolic metric on $\mathbb{C}*$ of Gauss' curvature −4.

One can require in this theorem much more, namely, that the desired quasiconformal automorphism ${\displaystyle \widehat{\mathbb{C}}}\to {\displaystyle \widehat{\mathbb{C}}}$ belongs to a given homotopic class of homeomorphisms of the punctured spheres $\mathbb{C}(a)\to \mathbb{C}(a^{\prime })$ . The different proofs and some applications of theorem can be found in [Ah2,Ag1,Ho1,Ho2,Kr3,KK,LVV], We provide here another application, following [Kru5].

It suffices to consider the quadruples (0, 1, α, ∞), applying additional fractional-linear transformations of ${\displaystyle \widehat{\mathbb{C}}}$ to the initial ones. Let ${\displaystyle \tilde{\mathbb{C}}}(\alpha )$ be the two-sheeted covering of ${\displaystyle \widehat{\mathbb{C}}}$ with branch points 0, 1, α, ∞; it is conformally equivalent to a torus X. This conformal isomorphism ${\displaystyle \widehat{\mathbb{C}}}(\alpha )\leftrightarrow X$ is realized by the elliptic integral of the first kind

(2.29) $u={\displaystyle {\int }_{z_0}^z\frac{dt}{\sqrt{t(t-1)(t-\alpha )}},}$

where z ₀ is a fixed point distinct from 0, 1, α and ∞, and a fixed branch of the square root in a neighborhood of z ₀ is chosen. We take a canonical dissection of ${\displaystyle \tilde{\mathbb{C}}}(\alpha )$ formed by the twice converted cuts along a Jordan arc γ ₁ connecting the points 0 and 1 and along a Jordan arc γ ₂ connecting the points 0 and α. The image of the dissected surface ${\displaystyle \tilde{\mathbb{C}}}(\alpha )$ under the map (2.29) is a topological quadrilateral G in the plane ${\mathbb{C}}_u$ with pairwise identified opposite sides. Its conformal modulus τ = ω ₂/ω ₁, where

${\omega }_j={\displaystyle {\int }_{{\gamma }_j}\frac{dz}{\sqrt{z(z-1)(z-\alpha )}}}.$

Now, let $a^{\prime }=(a_1^{\prime },a_2^{\prime },a_3^{\prime },a_4^{\prime })$ be another ordered quadruple; it is equivalent to (0, 1, α′, ∞). Any quasiconformal automorphism ${\displaystyle \widehat{\mathbb{C}}}\to {\displaystyle \widehat{\mathbb{C}}}$ moving a into a′ (i.e., with fixed points 0, 1, ∞ and moving α into α′) is lifted to a quasiconformal homeomorphism ${\displaystyle \tilde{f}}:{\displaystyle \tilde{\mathbb{C}}}(\alpha )\to {\displaystyle \tilde{\mathbb{C}}}({\alpha }^{\prime })$ with $K({\displaystyle \tilde{f}})=K(f)$ . Applying the conformal maps of both tori ${\displaystyle \tilde{\mathbb{C}}}(\alpha )$ and ${\displaystyle \tilde{\mathbb{C}}}({\alpha }^{\prime })$ one obtains that

(2.30) $q={\displaystyle \underset{f}{\min }}K(f)=\frac{|{\tau }^{\prime }-{\displaystyle \overline{\tau }}|+|{\tau }^{\prime }-\tau |}{|{\tau }^{\prime }-{\displaystyle \overline{\tau }}|-|{\tau }^{\prime }-\tau |}$

(where $2{\omega }_1^{\prime }$ and $2{\omega }_2^{\prime }$ are the corresponding periods for ${\displaystyle \tilde{\mathbb{C}}}({\alpha }^{\prime })$ and ${\tau }^{\prime }={\omega }_2^{\prime }/{\omega }_1^{\prime }$ . Hence, the extremal map f ₀ minimizing K(f) corresponds to the affine map

(2.31) $u^{\prime }=\frac{({\tau }^{\prime }-{\displaystyle \overline{\tau }})u+(\tau -{\tau }^{\prime }){\displaystyle \overline{u}}}{\tau -{\displaystyle \overline{\tau }}}$

of the plane ${\mathbb{C}}_u$ . Only these maps determine the boundary points of the non-Euclidean disk

$\Delta \left({\alpha }^{\prime }\right)=\left\{{\alpha }^{\prime }\in {\mathbb{C}}^{*}:{\rho }_{{\mathbb{C}}^{*}}\left(\alpha ,{\alpha }^{\prime }\right)=\frac{1}{2}\log q\right\}$

in Theorem 2.6.

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Functional Analysis and its Applications

Vladimir V. Kisil , in North-Holland Mathematics Studies, 2004

3 Representations of SL(2, ℝ) in Banach Algebras

A simple but important observation is that the Möbius transformations (4) can be easily extended to any Banach algebra. Let $\mathfrak{A}$ be a Banach algebra with the unit e, an element $a\in \mathfrak{A}$ with ||a|| < 1 be fixed, then

(7) $g:a\mapsto g\cdot a=\left(\overline{\alpha }a-\overline{\beta }e\right){\left(\alpha e-\beta a\right)}^{-1},g\in \text{SL}\left(2,ℝ\right)$

is a well defined SL(2, ℝ) action on a subset $\text{A}=\left\{g\cdot a|g\in \text{SL}\left(2,ℝ\right)\right\}\subset \text{A}$ , i.e. $\text{A}$ is a SL(2, ℝ)-homogeneous space. Let us define the resolvent function $R\left(g,a\right):\text{A}\to \text{A}$ by the familiar formula R(g, a)   =   (αe – βa)⁻¹ then

(8) $\begin{array}{l}{R}_1\left(g_1,\text{a}\right)R_1\left(g_2,g_1^{-1}\text{a}\right)=R_1\left(g_1{g}_2,\text{a}\right)\hfill \end{array}.$

The last identity is well known in representation theory [3, § 13.2(10)] and is a key ingredient of induced representations. Thus we can again linearise (7) (cf. (5)) in the space of continuous functions $C\left(\text{A},M\right)$ with values in a left $\text{A}\text{-module}$ M, e.g. $M=\text{A}:$

(9) $\begin{array}{l}{\rho }_a\left(g_1\right):f\left(g^{-1}\cdot a\right)\mapsto R\left(g_1^{-1}{g}^{-1},a\right)f\left(g_1^{-1}{g}^{-1}\cdot a\right)={\left({\alpha }^{\prime }e-{\beta }^{\prime }a\right)}^{-1}f\left(\frac{{\overline{\alpha }}^{\prime }\cdot a-{\overline{\beta }}^{\prime }e}{{\alpha }^{\prime }e-{\beta }^{\prime }a}\right)\hfill \end{array}.$

For any m ∈ M we can again define a K-invariant vacuum vector as $v_m\left(g^{-1}\cdot a\right)=m\otimes v_0\left(g^{-1}\cdot a\right)\in C\left(\text{A},M\right).$ It generates the associated with v_m family of coherent states v_m (u, a)   =   (ue – a)^−l m, where $u\in \text{D}$ .

The wavelet transform defined by the same common formula based on coherent states (cf. (6)):

${\mathcal{W}}_{m}f\left(g\right)=〈f,{\rho }_a\left(g\right)v_m〉,$

is a version of Cauchy integral, which maps $L_2\left(\text{A}\right)$ to C(SL(2, ℝ), M). It is closely related (but not identical!) to the Riesz-Dunford functional calculus: the traditional functional calculus is given by the case:

$\begin{array}{l}\text{Φ}:f\mapsto {\mathcal{W}}_{m}f\left(0\right)\text{for}M=\text{A}\text{and}m=e.\hfill \end{array}$

The both conditions—the intertwining property and initial value—required by Definition 5 easily follows from our construction.

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Geometric Function Theory

Samuel L. Krushkal , in Handbook of Complex Analysis, 2005

7.1.1

The universal Teichmüller space $\mathbb{T}$ is the space of quasisymmetric homeomorphisms h of the unit circle factorized by Möbius transformations. Its topology and real geometry are determined by the Teichmüller metric which naturally arises from extensions of those h to the unit disk. As was mentioned in Section 2.2, this space admits also the complex structure of a complex Banach manifold by means of the Bers embedding as a bounded subdomain of the Banach space $\mathbb{B}$ of holomorphic functions φ in the disk ∆* with the norm ||φ|| = sup _∆*(|z|² − 1)²|φ(z)|. Note that φ(z) = O(|z|⁻⁴) as z → ∞.

We shall identify the space $\mathbb{T}$ with this domain. In this model the points $\psi \in \mathbb{T}$ represent the Schwarzian derivatives S _f of univalent holomorphic functions f in ∆*, which have quasiconformal extensions to the whole sphere ${\displaystyle \widehat{ℂ}}$ .

Recall that the universal Teichmüller space $\mathbb{T}$ is obtained from the Banach ball

$\text{Belt}{(\Delta )}_1=\left\{\mu \in L_{\infty }(ℂ):\mu |\Delta *=0,||\mu ||<1\right\}$

of conformal structures on ${\displaystyle \widehat{ℂ}}$ by the natural identification, letting µ and v in Belt(∆)₁ be equivalent if w ^µ |S ¹ = w ^v |, S ¹, S ¹ = ∂∆. We denote the equivalence classes by [µ].

For an arbitrary (finitely or infinitely generated) Fuchsian group G with invariant unit circle ∂∆ we set

$\mathbb{B}(\Gamma )=\left\{\phi \in \mathbb{B}:(\phi \circ \gamma ){{\gamma }^{\prime }}^2=\phi \text{for}\text{all}\gamma \in \Gamma \right\},$

which is the space of hyperbolically bounded Γ-automorphic 2-forms. This yields that $\mathbb{T}$ contains the copies of Teichmüller spaces $\mathbb{T}(\Gamma )$ of arbitrary Riemann surfaces and of uniformizing Fuchsian groups. These spaces are isometrically embedded into $\mathbb{T}$ . It is established that $\mathbb{T}(\Gamma )=\mathbb{T}\cap \mathbb{B}(\Gamma )$ (see, e.g., [Leh2]). The spaces $\mathbb{T}(\Gamma )$ involve univalent holomorphic functions with quasiconformal extensions compatible with the Fuchsian and quasi-Fuchsian groups.

Let us introduce also the sets

$\begin{array}{ll}\mathbb{S}=\left\{\phi =S_f:f\text{univalent}\text{in}\Delta \text{*}\right\},\hfill & \mathbb{S}(\Gamma )=\mathbb{S}\cap \mathbb{B}(\Gamma ).\hfill \end{array}$

We consider on $\mathbb{S}(\Gamma )$ the topology induced by the norm in $\mathbb{B}$ ; the convergence in this topology is invariant with respect to Möbius transformations of ${\displaystyle \widehat{ℂ}}$ . The Schwarzian S _f can be regarded as a measure for deviation of the mapping f from a Möbius one.

In some instances in the sequel, it would be more convenient to consider the functions holomorphic in the disk ∆, instead of ∆*. We shall keep for this case the above notations.

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Geometric Function Theory

Frederick P. Gardiner , William J. Harvey , in Handbook of Complex Analysis, 2002

1.5 Tangent spaces to QS and S

In this section we identify the circle with the extended real line ${\displaystyle \overline{\mathbb{R}}}$ . By postcomposing with a Möbius transformation we may assume any homeomorphism representing an element of T fixes infinity. Consider a smooth curve h _s of homeomorphisms in QS parameterized by s and passing through the identity at s = 0. We may assume each homeomorphism h _s fixes infinity. By smooth we shall mean that

(4) $h_s(x)=x+sV(x)+\text{o}(s),$

where the distance measured in the quasisymmetric norm from the identity to h _s is less than or equal to a constant times s. In particular,

$\frac{1}{1+C_s}⩽\frac{h_s(x+t)-h_s(x)}{h_s(x)-h_s(x-t)}⩽1+Cs.$

By substituting (4) into this formula we arrive at the following condition on the continuous function V:

(5) $\left|V(x+t)-2V(x)+V(x-t)\right|=\text{O}(t).$

If h _s is a smooth curve in the symmetric subspace S, then

(6) $\left|V(x+t)-2V(x)+V(x-t)\right|=\text{o}(t).$

We will call (5) and (6), respectively, the big and little Zygmund conditions. Since V is to be regarded as the tangent vector to the one-parameter family of homeomorphisms h _s , $V(x)\frac{\partial }{\partial x}$ is a vector field.

If instead we consider the mappings h _s as acting on the unit circle |z| = 1 then the condition that the vector field W point in a direction tangent to the unit circle is that

(7) ${\displaystyle \tilde{W}}(x)=W\left(e^{\text{i}x}\right)/{\text{ie}}^{\text{i}x}$

be real-valued. The boundedness conditions on QS and S correspond to the conditions that the continuous, periodic function ${\displaystyle \tilde{W}}$ satisfy (5) and (6). We denote the spaces of continuous vector fields satisfying these conditions by Z and Z ₀, respectively.

A simple example of a tangent vector in Z ₀ is generated by a curve of Möbius transformations preserving the unit circle and passing through the identity. Such a curve has a tangent vector of the form

$W(z)\frac{\partial }{\partial z}=\left(\alpha z^2+\beta z+\gamma \right)\frac{\partial }{\partial z},$

where α, β and γ are constants which make W(z) real-valued along $z{\displaystyle \overline{z}}=1$ . We call such tangent vectors trivial. Thus the quadratic polynomials which satisfy this reality condition define the tangent vectors to trivial curves of homeomorphisms.

We will show that any tangent vector satisfying the big Zygmund condition is the tangent vector to a smooth curve in QS passing through the identity, and correspondingly any tangent vector satisfying the little Zygmund condition is the tangent vector to a smooth curve in S.

Definition. Let $\mathcal{Z}$ and ${\mathcal{Z}}_0$ be the spaces Z and Z ₀ factored by the quadratic polynomials.

Eventually in Section 2.3 we shall identify a Banach space $\mathcal{A}$ such that the Banach dual of ${\mathcal{Z}}_0$ is isomorphic to $\mathcal{A}$ and the Banach dual of $\mathcal{A}$ is isomorphic to $\mathcal{Z}$ . In particular, $Z_0^{**}\cong Z$ .

Let Q be any quadruple of points a,b,c and d arranged in counter-clockwise order on the unit circle or in increasing order on the real axis and define the cross ratio cr(Q) by

(8) $cr(Q)=\frac{(d-c)(b-a)}{(c-b)(a-d)}.$

Recall that cr(Q) is Möbius invariant in the sense that cr(A(q)) = cr(Q) for any Möbius transformation A. In consequence we may define a norm || || _cr on vector fields which is Möbius invariant in the sense that

$||W|{|}_{cr}=|\left|\frac{W\circ A}{A^{\prime }}\right|{|}_{cr},$

for every Möbius transformation A.

Define W[a, b, c, d] to be the alternating sum

$\frac{W(d)-W(c)}{d-c}-\frac{W(c)-W(b)}{c-b}+\frac{W(b)-W(a)}{b-a}-\frac{W(a)-W(d)}{a-d}.$

For a given quadruple Q the term cr(Q)ρ(cr(Q)) W[a, b, c, d] measures the velocity of the cross-ratio (8) with respect to the Poincaré metric ρ(z)|dz| on the sphere punctured at 0, 1 and ∞ when each of the points a, b, c and d move with complex velocities W(a), W(b), W(c) and W(d), respectively. The infinitesimal cross-ratio norm is defined for the space $\mathcal{Z}$ by

(9) $||W|{|}_{cr}={\displaystyle \underset{Q}{\sup }}|cr(Q)\rho \left(cr(Q)\right)W[a,b,c,d]|.$

Note that ||W|| _cr = 0 if, and only if, W is a quadratic polynomial. Furthermore, if Q has the form Q = (−∞, x − t, x, x + t) then cr(Q) = −1. If in addition we assume |W(z)| = o(|z|²), which is tantamount to the assumption that $W(z)\frac{\partial }{\partial z}$ vanishes at infinity, then the alternating sum W[a, b, c, d] is equal to

$-\frac{W(x+t)-2W(x)+W(x-t)}{t}.$

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Handbook of Dynamical Systems

Pascal Hubert , Thomas A. Schmidt , in Handbook of Dynamical Systems, 2006

2.3.3 Projecting Orbits to

M_g and A_g The projection $\pi :\Omega M_g\to M_g$ is constant on orbits of SO(2, ℝ). On the other hand, the stabilizer of z =i under the transitive action of SL(2, ℝ) by Möbius transformations on the Poincaré upper half-plane, ℍ, is SO(2, ℝ). There is thus a map ℍ→M_g that factors through SL(X, ω)\ℍ. In fact, it is of great importance that this image in M_g is isometrically immersed with respect to the so-called Teichmüller metric, see [13] for discussion of this metric in terms related to SL(2, ℝ). The image in M_g is an algebraic curve if and only if SL(X, ω) is a lattice, in which case this image is called a Teichmüller curve in M_g .

The Torelli map $\tau :M_g\to A_g$ is defined by sending each X to Jac(X); for a discussion of the geometry of this map, see [41]. In dimension g = 2, in fact $A_2=\tau (M_2)\bigsqcup H_1$ , where H ₁ is the locus of Abelian varieties that split as a product of two polarized elliptic curves. In particular, the Torelli map has dense open image in A ₂; there is thus a tendency in the literature to slur over the distinction of certain loci as being in one or the other of the spaces M ₂ and A ₂. For simplicity, call the map $\Omega M_g\to A_g$ , given by composing the Torelli map with π, the projection to A _g .

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Geometric Function Theory

R. Kühnau , in Handbook of Complex Analysis, 2005

1 Introduction

Let a fixed domain G be given in the complex z-plane or on the Riemann sphere. With the exception of Section 7 "domain" always means a finitely-connected domain. A fundamental question is then: Are there in the class of all schlicht conformal mappings of G special mappings which are in some sense distinguished? That means the question for mappings with some special analytic or mainly geometric properties. In the latter case we seek mappings which produce image domains with some prescribed geometric shape, so called "canonical" or "representative" domains. The question for canonical conformal mappings consists in the question of the existence and in the question of uniqueness by setting of side conditions, so called normalizations. Such normalizations are necessary for uniqueness because we have otherwise some freedom with the use of Möbius transformations. Because Möbius transformations depend on 3 complex numbers, also these normalizations consist in the simplest case in 3 side conditions.

If we set more than 3 normalizations ("höhere Normierungen" in the terminology of Grötzsch [26]), the things become more complicated, also in the case we consider G and its images as a part of a fixed Riemann surface.

We will see that there is a very rich theory of canonical conformal mappings with many aspects, and connections to many other fields and questions: Identities between these mappings and connections with fundamental solutions and kernel functions, extremal problems to characterize these canonical mappings, etc.

In the (mainly considered) conformal case there exists a rich literature about canonical mappings. We mention mainly the books [9,17,23,29,32,35,82,97,99,110] and the article [18]. Therefore we will give here only some typical examples and will then restrict us to sketch some additional aspects and especially to the generalization of the theory to conformal mappings with quasiconformal extension.

What concerns the main methods for proving the existence of canonical conformal mappings we mention: methods for solving boundary value problems, extremal problems combined with normal family arguments (the most elegant method, e.g., in the case of the parallel slit mapping), Koebe's method of continuity, functional-analytic fix-point methods, orthonormal expansions, integral equations.

To begin with let us start with the simplest case of a simply-connected domain G on the Riemann sphere. If G has no or only one boundary point then the situation is trivial because the Möbius transformations are the only schlicht conformal mappings of G. If G has more than one boundary point we have the fundamental Riemann mapping theorem: For every such G there exists a schlicht conformal mapping onto the unit disk. The mapping is unique up to a following Möbius transformation of the unit disk onto itself. These Möbius transformations contain 3 real parameters.

We can find the Riemann mapping theorem in almost all textbooks on Complex Analysis (cf. also references in [70]), with all corresponding aspects as the connection with the Green's function of G, construction of the Riemann mapping for polygonal domains G (Schwarz–Christoffel formula), construction with orthogonal expansions etc. We can find the boundary behavior of the Riemann mapping, e.g., in [90,91]. For numerical procedures cf. [17] and [112] with many references.

From the Riemann mapping theorem it follows that all simply-connected domains with more than one boundary point are conformally equivalent. If we pass now over to doubly-connected domains G we have a new situation: Two such domains are conformally equivalent if and only if the so-called conformal module is the same. This means in particular that every doubly-connected domain G is conformally equivalent to an annulus with the same conformal module. (This annulus can degenerate.) One can find a detailed discussion of the corresponding questions in [69]. We will restrict ourselves here therefore to the general case of conformal mappings of multiply-connected domains, including the case of connectivity greater than 2.

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Geometries for CAGD

Helmut Pottmann , Stefan Leopoldseder , in Handbook of Computer Aided Geometric Design, 2002

3.2.2 Möbius geometry

Let Eⁿ be real Euclidean n-space, P its point set and M the set of (non-oriented) hyperspheres and hyperplanes of Eⁿ. We obtain the so-called Euclidean conformal closure Eⁿ _M of Eⁿ by extending the point set P by an arbitrary element (ideal point) ∞ ∉ P to P_M = P ∪ {∞}. As an extension of the incidence relation we define that ∞ lies in all hyperplanes but in none of the hyperspheres. The elements of M. are called Euclidean Möbius hyperspheres.

Euclidean Möbius geometry is the study of properties that are invariant under Euclidean Möbius transformations. A Möbius transformation is a bijective map of P_M , which maps Möbius hyperspheres to Möbius hyperspheres. A simple example is given by the inversion x ↦ $\frac{r^2}{x^2}$ with respect to the sphere x ² = r ² in ℝ ⁿ . Another example is the reflection at a hyperplane, viewed as Möbius sphere. Any general Möbius transformation is a composition of inversions with respect to Möbius spheres.

Besides the standard model of Euclidean Möbius geometry, mentioned above, we obtain the quadric model of this geometry by embedding Eⁿ in Euclidean n + 1-space E^{n +1} as plane x_{n +1} = 0. Let σ: ∑ \{z} → Eⁿ be the stereographic projection of the unit hypersphere

(3.10) ${{\displaystyle \sum :x_1^2+\mathrm{\dots }+x}}_{n=1}^2=1$

onto Eⁿ with projection center (or north pole) z = (0,…, 0, 1), see Figure 3.7.

Figure 3.7. Stereographic projection, top and front view.

Extending σ to $\overline{\sigma }$ with $\overline{\sigma }$ : z↦∞gives the quadric model of Euclidean Möbius geometry which is related to the standard model via $\overline{\sigma }$ . The point set is that of ∑ ⊂ E ^{n +1} and the Möbius spheres are the hyperplanar intersections of ∑ since σ is preserving hyperspheres.

For the analytic treatment of Euclidean Möbius geometry, let $\tilde{x}-\left({\tilde{x}}_1,\mathrm{\dots },{\tilde{x}}_n\right)$ denote a point in Eⁿ , and $x={\sigma }^{-1}\left(\tilde{x}\right)=\left(x_1,\mathrm{\dots },x_{n+1}\right)$ the corresponding point of ∑ ⊂ E^{n +1}. Let P^{n +1} denote the projective extension of E^{n +1}. In homogeneous coordinates we then have x ℝ = (x ₀, x ₁,…, x_{n +1})ℝ with -x ² ₀ + x ² ₁ + … + x ² _n+1 = 0 (x ℝ ∈ ∑). The inverse stereographic projection σ⁻¹: P → ∑\{z} ⊂ E^{n +1} is given by

(3.11) ${\sigma }^{-1}\left(\tilde{\text{x}}\right)=\times \mathcal{R}=\left({\tilde{x}}_1^2+\mathrm{\dots }+{\tilde{x}}_n+1,2{\tilde{x}}_1,2{\tilde{x}}_n,\mathrm{\dots },2{\tilde{x}}_n,{\tilde{x}}_1^2+\mathrm{\dots }+{\tilde{x}}_n-1\right)\mathcal{R}.$

The homogeneous coordinates x = (x ₀,x ₁,…,x_{n +1}) are called n-spherical coordinates of a point $\tilde{x}\in {\text{E}}^n$ . These coordinates are appropriate to represent Möbius spheres as well: Via σ⁻¹ a Möbius sphere M ∈ M corresponds to a hyperplanar intersection of ∑, whose pole with respect to ∑ shall be denoted by c ℝ, see Figure 3.7. Its homogeneous coordinates

$c=\left(c_0,c_1,\mathrm{\dots },c_{n+1}\right)$

are called the n-spherical coordinates of M. For n = 2,3 these coordinates are usually denoted by tetracyclic and pentaspherical coordinates, respectively.

It can be easily verified that in case of c ₀ = c_{n +1} the Möbius sphere M represents a hyperplane of the standard model with equation -c ₀ + c ₁ x ₁ + … + c_{n +1} x_{n +1} = 0. In case of c ₀ ≠ c_{n +1} the Möbius sphere M represents the hypersphere with midpoint 1/(c ₀ - c_{n +1}) ˙ (c ₁,…,c_n ) and radius (c ² ₁ + … + c ² _n+1 - c ² ₀)/(c ₀ - c_{n +1})². Let

${〈x,y〉}_M=-{\text{X}}_0{y}_0+x_1{y}_1+\mathrm{\dots }+{\text{X}}_{n+1}{y}_{n+1}={\times }^{\text{T}}.{\text{E}}_M.y$

with E _M = diag(-1, 1,…, 1) describe an indefinite scalar product. Then we are able to describe points by n-spherical coordinates x with 〈x, x〉 _M = 0 and Möbius spheres by n-spherical coordinates c with 〈c, c〉 _M > 0. Incidence of a point x ℝ and a Möbius sphere c ℝ is given by 〈x, c〉 _M = 0.

It is a central theorem of Euclidean Möbius geometry that in the quadric model all Euclidean Möbius transformations are induced by linear maps P ^{n +1} → P ^{n +1},x ↦ A ˙ x with A ^T ˙ E _M ˙ A = λE _M , where P ^{n +1} again denotes the projective extension of E ^{n +1}. These linear maps represent those projective maps of P ^{n +1} that keep ∑ fixed (as a whole).

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Geometric Function Theory

Kenneth Stephenson , in Handbook of Complex Analysis, 2002

1.2 Maximal packings

Given a complex $\mathcal{K}$ , there are numerous results on the existence and variety of packings for $\mathcal{K}$ ; we need two main types. Let us begin with the extreme rigidity displayed by maximal packings.

Circle Packing Theorem

Given a complex $\mathcal{K}$ , there exists a unique Riemann surface $\mathcal{X}$ and a circle packing $P_{\mathcal{K}}$ ; for $\mathcal{K}$ . in the intrinsic metric of constant curvature on $\mathcal{X}$ so that $P_{\mathcal{K}}$ is univalent and fills $\mathcal{X}$ . The packing $P_{\mathcal{K}}$ is unique up to conformal automorphisms of $\mathcal{X}$ and is called the maximal packing for $\mathcal{K}$ .

This is just the Koebe–Andreev–Thurston Theorem when $\mathcal{K}$ triangulates a sphere; it was extended by Beardon and Stephenson to arbitrary $\mathcal{K}$ . of bounded degree [10] and then by He and Schramm to cases of unbounded degree [55].

Figure 4 illustrates maximal packings for several complexes. Note that each $\mathcal{K}$ "chooses" its own geometry: that is, $\mathcal{K}$ , which begins as a topological surface, is endowed with a unique conformal structure and associated metric which support $P_{\mathcal{K}}$ . The complex $\mathcal{K}$ , is said to be spherical, parabolic, or hyperbolic depending on the curvature of that geometry.

Fig. 4. Maximal packings

We cannot go into the details of the proof, but it may be helpful to highlight some of the familiar geometric notions which are pivotal. The first case is the key, both in theory and practice.

Case I $\mathcal{K}$ , triangulates a closed topological disc

Here $\mathcal{K}$ . is finite, simply connected, and has a boundary. Figure 2(a) is an elementary example, and the extremal nature of its maximal packing, Figure 2(d), is visually evident – the boundary circles are horocycles, circles internally tangent to the unit circle. Indeed, horocycles are naturally interpreted as circles of infinite hyperbolic radius, so our task reduces to finding a hyperbolic packing label for $\mathcal{K}$ whose boundary labels are infinite.

Formally, the proof relies on monotonicity results and a "Perron" method. One defines Ф as the collection of subpacking labels for $\mathcal{K}$ , that is, (hyperbolic) labels R with the property that the associated angle sums for interior vertices are greater than or equal to 2π. One shows that Φ is non-empty, closed under maxima, and uniformly bounded for interior v. The upper envelope, the label ${\displaystyle \widehat{R}}$ defined by

${\displaystyle \widehat{R}}(v)={\displaystyle \underset{R\in \Phi }{\sup }}\left\{R(v)\right\},$

can be shown to be a packing label with infinite boundary radii. With these radii in hand, successively laying out the circles gives $P_{\mathcal{K}}$ .

It is more informative (and practical) to approach the packing problem algorithmically. The geometric ingredients are surprisingly elementary. Consider a univalent flower in $\mathbb{D}$ as shown below, with central (hyperbolic) radius r and m petal radii r ₁, r ₂,…, r _m ; the (hyperbolic) faces are shown for reference. Then:

Monotonicity: If r is increased, then the angle sum at the center circle goes down while the angle sum at each of the petal circles goes up.

Rodin/Sullivan Ring Lemma: There's a constant C = C(m) so that r _j /r⩾ C; that is, in a (univalent) flower, no petal circle can be too much smaller than the center circle.

Bound: r ⩽ − log(sin(π/m)). In the hyperbolic plane, a circle cannot be too large if m petal circles can wrap at least once around it.

To get some taste for the reasoning, suppose we were to grant the existence of some initial hyperbolic packing P ₀ for $\mathcal{K}$ . I ask the reader to imagine the effect of increasing one of its boundary circles. It's an interesting exercise depending solely on monotonicity; deduce an upward pressure reverberating through all of the interior radii as they adjust in order to keep their angles sums at 2π. The adjustments ultimately lead to a new packing for $\mathcal{K}$ . which accommodates the increased boundary circle. Iterating this increment/repack cycle allows one to push the boundary radii to ∞; monotonicity and our bound force the interior radii to converge to finite limiting values. The result is the maximal packing label $R_{\mathcal{K}}$ . The argument is quite striking when implemented live on a computer screen. Figure 5(a) is an initial packing P ₀ for the complex of Figure 2. Incrementally increasing the boundary radii generates a succession of intermediate packings, several of which are superimposed in Figure 5(b). One can see the monotonicity at work as the maximal packing, isolated in Figure 5(c), emerges.

Fig. 5. Growing to the max.

The bound on radii for interior circles, which fails in euclidean geometry, along with the availability of circles (horocycles) having infinite radius are what make the proof click in the hyperbolic setting. The uniqueness of $P_{\mathcal{K}}$ (up to automorphisms of $\mathbb{D}$ ) can be established using monotonicity along with the association in hyperbolic geometry between "angles" and "area". (It also follows from fixed point arguments due to He and Schramm, which we'll comment on later.) The fact that the circles of the maximal packing have mutually disjoint interiors is a consequence of local univalence and a standard topological argument principle.

Let's register an important observation about $R_{\mathcal{K}}$ which can be spun off from the proof.

Lemma 2

Suppose $\mathcal{K}$ triangulates a closed topological disc. If R is any hyperbolic packing label for $\mathcal{K}$ , then $R⩽R_{\mathcal{K}}$ , meaning that $R(v)⩽R_{\mathcal{K}}(v)$ for every vertex $v\in \mathcal{K}$ . Moreover, equality holds at a single interior vertex if and only if $R=R_{\mathcal{K}}$ .

This justifies the adjective "maximal" for $P_{\mathcal{K}}$ and will shortly become the key ingredient for handling infinite complexes.

Case II $\mathcal{K}$ triangulates a sphere

The spherical case is an easy consequence of Case I. Choosing a vertex v _∞ of $\mathcal{K}$ , form a reduced complex ${\mathcal{K}}^{\prime }$ by removing v _∞ and all edges and faces containing v _∞. Then ${\mathcal{K}}^{\prime }$ triangulates a closed disc, so Case I yields a maximal packing $P^{\prime }=P_{{\mathcal{K}}^{\prime }}$ in $\mathbb{D}$ . Stereographic projection to ${\mathbb{S}}^2$ carries $P^{\prime }$ to a circle packing in the southern hemisphere. All its boundary circles, as projections of horocycles in $\mathbb{D}$ , will be tangent to the northern hemisphere, so by including that hemisphere as the circle c _∞ for v _∞, we have a spherical packing in ${\mathbb{S}}^2$ for $\mathcal{K}$ itself. A Möbius transformation can be applied for any desired normalization.

This stereographic projection argument is Thurston's and was originally used to prove Case I from Case II. However, as you will see in the next case, the hyperbolic arguments give added value.

Case III $\mathcal{K}$ triangulates an open topological disc

Here $\mathcal{K}$ is infinite, simply connected, and without boundary. Designate some interior vertex v ₀ and exhaust $\mathcal{K}$ , by a nested sequence of finite simply connected complexes $v_0\in {\mathcal{K}}_1\subset {\mathcal{K}}_2\subset \cdots \to \mathcal{K}$ . For each j, write P _j for the maximal packing of ${\mathcal{K}}_j,R_j$ for the maximal packing label, and $r_0^{(j)}$ for the label at v ₀. Note that within the maximal packing P _j+1 lies a packing for ${\mathcal{K}}_j$ . By Lemma 2, therefore, $r_0^{(j+1)}<r_0^{(j)}$ ; that is, the labels at v ₀ are strictly decreasing,

$r_0^{(1)}>r_0^{(2)}>r_0^{(3)}>\cdots .$

Fig. 6. Projection to the sphere.

In other words, as more circles of $\mathcal{K}$ are incorporated, the circle at the origin gets smaller. That brings us to a crucial

$\begin{array}{cc}\text{Dichotomy:}& \{\begin{array}{ll}(\text{P}):\hfill & r_0^{(j)}\to 0,\hfill \\ (\text{H}):\hfill & r_0^{(j)}\to r_0^{(\infty )}\text{for}\text{some}{r}_0^{(\infty )}>0.\hfill \end{array}\end{array}$

We will find that alternatives (P) and (H) correspond to parabolic and hyperbolic, respectively. It might be best to illustrate with a pair of particularly clean examples. Let ${\mathcal{K}}^6$ denote the familiar hexagonal (constant 6-degree) complex and ${\mathcal{K}}^7$ , the heptagonal (constant 7-degree) complex. In each instance choose ${\mathcal{K}}_j$ to consist of vertices within j-generations of v ₀ and normalize P _j so that v ₀ is at the origin and some designated neighbor v ₁ is centered on the positive imaginary axis.

In Figure 7(a) I have superimposed the first few packings P _j associated with ${\mathcal{K}}^6$ . Note that the circle for v ₀ at the origin is shrinking rather rapidly – this is alternative (P). In contrast, in Figure 7(b) I have superimposed the first few packings associated with ${\mathcal{K}}^7$ ; the circle at the origin again gets smaller, but very quickly stabilizes at a positive value – this is alternative (H).

Fig. 7. The hyperbolic/parabolic dichotomy.

The maximal packing $P_{\mathcal{K}}$ in each case results from a "geometric" limiting process. In the heptagonal case, Figure 7(b), one can almost see the limit packing emerging. Formally, one uses the Ring Lemma to prove existence of positive limit radii for every vertex v and diagonalization to deduce limits for circle centers. It is relatively easy to show that the limit packing P is univalent and fills $\mathbb{D}$ . In other words, ${\mathcal{K}}^7$ is hyperbolic and the packings P _j converge to $P_{{\mathcal{K}}^7}$ .

On the other hand, it is certainly difficult to see a penny packing emerging in Figure 7(a). Indeed, in alternative (P), the Ring Lemma implies that all circle radii decrease to zero. Consequently, we shift perspective, treating each P _j as a euclidean packing and scaling it so the circle for v ₀ is the unit circle. Now the Ring Lemma yields positive and finite limits for all the (euclidean) radii and again diagonalization provides us with a univalent limit packing P. Thus ${\mathcal{K}}^6$ is parabolic and $P=P_{{\mathcal{K}}^6}$ , the familiar "penny" packing. The proof that carr $P_{{\mathcal{K}}^6}=\mathbb{C}$ in the parabolic case is more difficult than it might seem. It was confirmed for $\mathcal{K}$ having bounded degree in [10] using quasiconformal arguments. The proof in the general case is even more subtle and was provided by He and Schramm in [55]. Their key observation? Distinct circles can intersect in at most 2 points! I can't give details, but their arguments deserve mention not only for their elegance but for the powerful tools they bring to the discrete setting – versions of the winding number arguments so central in classical complex analysis.

If $\mathcal{K}$ is infinite, simply connected, but has boundary, then it falls under the hyperbolic alternative (H). With this observation, we find that we have taken care of all simply connected complexes, Case I being hyperbolic, Case II spherical, and Case III either hyperbolic or parabolic depending on combinatorics. Before going on, it is important to note the essential uniqueness of all the extremal packings obtained so far: when $\mathcal{K}$ is simply connected, its maximal packing $P_{\mathcal{K}}$ is unique up to Möbius transformations of the sphere, plane, or disc, as appropriate.

Case IV $\mathcal{K}$ triangulates a surface S

We assume that S is an oriented topological surface and, in view of our earlier cases, that S is not simply connected. It is well known that a triangulation $\mathcal{K}$ of S can be lifted to a complex ${\displaystyle \tilde{\mathcal{K}}}$ triangulating the universal covering surface ${\displaystyle \tilde{S}}$ of S. There is an associated simplicial projection $p:{\displaystyle \tilde{\mathcal{K}}}\to \mathcal{K}$ and a group G of simplicial automorphisms $g:{\displaystyle \tilde{\mathcal{K}}}\to \mathcal{K}$ , satisfying p ∘ g ≡ p.

${\displaystyle \tilde{\mathcal{K}}}$ is simply connected, and since the sphere covers only itself, ${\displaystyle \tilde{\mathcal{K}}}$ must be an infinite triangulation of a topological disc, Case III. Let $\mathcal{D}$ denote the plane or disc, depending on whether ${\displaystyle \tilde{\mathcal{K}}}$ is parabolic or hyperbolic, and let ${\displaystyle \tilde{P}}=P_{{\displaystyle \tilde{\mathcal{K}}}}$ denote the maximal packing in $\mathcal{D}$ for ${\displaystyle \tilde{\mathcal{K}}}$ . The situation is illustrated in Figure 8 for a hyperbolic case. The circle packing shown in $\mathbb{D}$ is just the part of the infinite packing ${\displaystyle \tilde{P}}$ associated with a fundamental domain for the covering p.

Fig. 8. Discrete covering map.

The essential uniqueness of ${\displaystyle \tilde{P}}$ in $\mathcal{D}$ becomes the key ingredient as we deploy a standard arrow-chasing argument. Briefly, each simplicial automorphism g of ${\displaystyle \tilde{\mathcal{K}}}$ must induce a Möbius transformation M _g of $\mathcal{D}$ which maps the packing ${\displaystyle \tilde{P}}$ to itself. Γ = {M _g : g ∈ G} is a discrete group of Möbius transformations of $\mathcal{D}$ isomorphic to G. Let $\mathcal{X}$ denote the Riemann surface $\mathcal{D}/\Gamma$ obtained in the classical manner from $\mathcal{D}$ by identifying all points equivalent modulo Γ, and write $\pi :\mathcal{D}\to \mathcal{X}$ for the analytic covering projection. As topological surfaces, $\mathcal{X}$ and S are homeomorphic, however $\mathcal{X}$ inherits a conformal structure and conformal metric from $\mathcal{D}$ under π. This "intrinsic" metric is either hyperbolic or euclidean, depending on $\mathcal{D}$ , and each circle in $\mathcal{D}$ projects, a fortiori, to a "circle" in the intrinsic metric on $\mathcal{X}$ . Clearly, the projected circles $\pi ({\displaystyle \tilde{P}})$ in $\mathcal{X}$ provide an in situ packing for $\mathcal{K}$ . This is precisely the maximal packing $P_{\mathcal{K}}$ we have been looking for.

This concludes our overview of maximal packings. Even in this last case, note that $\mathcal{X}$ is uniquely determined among all Riemann surfaces homeomorphic to S based purely on the combinatorics of $\mathcal{K}$ , so the take-home message is that $\mathcal{K}$ again "chooses" the appropriate geometry for its maximal packing.

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