# Similarity surfaces, connections, and the measurable Riemann mapping theorem

Arnaud Chéritat  and  Guillaume Tahar

###### Abstract.

This article studies a particular process that approximates solutions of the
Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann
mapping theorem) on ${\mathbb{C}}$. It passes through the introduction of a
sequence of similarity surfaces constructed by gluing polygons, and we explain
the relation between their conformal uniformization and the Schwarz-
Christoffel formula. Numerical aspects, in particular the efficiency of the
process, are not studied, but we draw interesting theoretical consequences.
First, we give an independent proof of the analytic dependence, on the data
(the Beltrami form), of the solution of the Beltrami equation (Ahlfors-Bers
theorem). For this we prove, without using the Ahlfors-Bers theorem, the
holomorphic dependence, with respect to the polygons, of the Christoffel
symbol appearing in the Schwarz-Christoffel formula. Second, these Christoffel
symbols define a sequence of parallel transports on the range, and in the case
of a data that is $C^{2}$ with compact support, we prove that it converges to
the parallel transport associated to a particular affine connection, which we
identify.

###### Contents

  1. Introduction
  2. 2 Polygons
  3. 3 The Schwarz-Christoffel formula
  4. 4 Holomorphic dependence
  5. 8 On averaging the Beltrami form

## Introduction

More than fifty years later, Chapter II of [Ahl66] remains an excellent
introduction to the theory of quasiconformal mappings and we highly recommend
it.111Or the second edition [Ahl06], which was issued in 2006. This book also
includes in Chapter V a proof of the measurable Riemann mapping theorem (see
LABEL:sec:mrmt here for a statement). This proof makes use of integral
operators with singular kernels (Ahlfors-Beurling operator), a good dose of
$L^{p}$ spaces for $p>2$, letting $p\longrightarrow 2$, and a subtle
inequality due to Calderón and Zygmund. There has been prior and posterior
proofs with different approaches in special cases or in the general case,
depending on the data $\mu$ in the statement (the article [Glu08] gives a
short historical overview and references):

  * $-$

Gauss (1825) for an ${\mathbb{R}}$-analytic $\mu$, by complexifying
${\mathbb{R}}^{2}$ into ${\mathbb{C}}^{2}$ and using a clever trick that does
not extend to the $C^{\infty}$ class;

  * $-$

Korn (1914) and Lichtenstein (1916) for Hölder-continuous $\mu$;
Lichtenstein’s method already involves integral operators. Korn uses a fixed
point method involving solving the Laplacian;

  * $-$

Lavrentiev (1935) for a continuous $\mu$, by constructing approximations to
the solution using conformal geometric methods (we sum-up the method in
LABEL:part:beltrami) [Lav35, Lav20];

  * $-$

Morrey (1936) for a general, measurable, $\mu$, using a density argument to
reduce to solving the case where $\mu$ is analytic;

to cite only a few. According to [Boj58] (translated in [Boj09]), the idea to
introduce the Ahlfors-Beurling operator (it did not bear that name at that
time) for this problem is due to Vekua [Vek55]. Concerning the holomorphic
dependence on the data $\mu$, all the proofs that the authors of the present
article know use the Ahlfors-Beurling operator. These works include:

  * $-$

Ahlfors and Bers, via $L^{p}$ spaces

  * $-$

Buff-Douady, via $L^{2}$ spaces only and the Fourier transformation

  * $-$

Glutsyuk, via a limit of a version for the torus, which is proved using
Fourier series.

LABEL:part:beltrami in the present article is an addition to this list, with
the difference that we do not use the Ahlfors-Beurling operator. Our proof
uses distributions and $L^{2}$ spaces for the definition of quasiconformal
maps, a classical compactness argument via an $L^{2}$ estimate, a new
ingredient that is similarity surfaces, and the Poincaré-Koebe theorem.222… Of
which there are several proofs that do not use Beltrami forms. See for
instance Chapter 10 of [Ahl10], or [Hub06].. It bears resemblance with the
approach of Lavrentiev and in fact the article of Lavrentiev avoids the use of
the Poincaré-Koebe theorem so it is not impossible that the proof we present
may be adapted too to avoid this use, but this seems not obvious.

Structure and content of the article

LABEL:part:simsurf is an introduction to the subject of similarity surfaces
with a focus on ones that are conformally equivalent to punctured Riemann
spheres and obtained by gluing finitely many convex polygons, possibly
unbounded (Section 2). In Section 3 we draw a link with a generalization of
the Schwarz-Christoffel formula: it expresses straightening charts. We believe
this fact was known before but we do not have references for this. In the
formula a particular rational map appears, which is the Christoffel symbol
(expression in a chart) of a particular conformal connection. Section 4 is
devoted to a proof of holomorphic dependence of the rational map appearing as
a Christoffel symbol, when the polygons that are glued are modified by letting
the vertices vary holomorphically. A key point is to completely avoid using
the measurable Riemann mapping theorem in this proof. Such an insistance
induces complications that are dealt with in Section 4.4.

LABEL:part:beltrami states the measurable Riemann mapping theorem (solution of
the Beltrami equation associated to a Beltrami form $\mu$) and details our
proof. A crucial point is the holomorphic dependence, which relies on the
holomorphic dependence of the Christoffel symbol proved in LABEL:part:simsurf.
We first describe an (already known) density method with some amount of
generality. Then we apply the results of the previous section to finalize the
proof.

The density method does not involve the Ahlfors-Beurling operator nor $L^{p}$
spaces beyond $L^{1}$, $L^{2}$. It involves the notion of distribution and the
Sobolev space $W^{1,2}_{\operatorname{loc}}$, which serve in particular for
the definition of quasiconformal maps. It also involves compactness statements
for quasiconformal maps: one about normal families, which is proved in [Ahl66]
by conformal geometry techniques; one about $L^{2}$ bounds, which is proved in
[Ahl66] by proving differentiability almost everywhere and bounding the
differential with the Jacobian.

The presentation of our proof is not completely self contained: part of the
book of Ahlfors, [Ahl66, Ahl06], is considered as a basic reference. In
particular several lemmas of this book are used here without proof, the
interested reader will have to check such proofs there.333None of these proofs
uses the Ahlfors-Beurling integral operator. Details about others aspects,
proofs, or prerequisite knowledge are given in appendices here.

While LABEL:part:beltrami presents an original approach to known results,
LABEL:part:cnx is prospective and contains new results. In
LABEL:part:beltrami, the solution $f$ of the Beltrami equation in the
measurable Riemann mapping theorem is obtained as a limit of approximations
$f_{n}$. Each approximation is associated to a particular abstract similarity
surface $\cal S_{n}$, with $(2n^{2}+1)^{2}+1$ singularities. Uniformizing the
underlying Riemann surfaces to the Riemann sphere gives rise to a meromorphic
conformal connection on ${\mathbb{C}}$, whose associated Christoffel symbol
$\zeta_{n}$ is a rational map with $(2n^{2}+1)^{2}$ poles. A natural question
arises: is there a limit to these similarity surfaces? In the particular case
where the Beltrami form $\mu$ is $C^{2}$ with compact support, we state and
prove limit theorems concerning the sequence of connections $\zeta_{n}$. In
particular, it tends in some sense to a conformal connection, but whose
Christoffel symbol $\zeta$, which we characterize, is not anymore a
holomorphic function. In some sense, we gave an answer to the above question:
the limit of the similarity surfaces $\cal S_{n}$ is the plane endowed with
the connection $\zeta$.

Part I

→

S

and a germ of chart $\tilde{s}_{0}$ at $\gamma(0)$, there is a unique way to
follow the germ along $\gamma$ so that for all $t$, $\tilde{s}_{t}$ is a germ
of chart at $\gamma(t)$ and so that the germs locally match,666This can be
expressed by saying that $t\mapsto\tilde{s}_{t}$ is continuous, for an
appropriate topology on the space of germs. i.e. for all $t$, $\exists\eta>0$
and a representative $s_{t}$ of $\tilde{s}_{t}$ such that for
$|t-t^{\prime}|<\eta$, $\gamma(t^{\prime})$ is in the domain of $s_{t}$ and
$\tilde{s}_{t^{\prime}}$ is the germ of $s_{t}$ at $\gamma(t^{\prime})$.
Existence can be proved by using an open cover of the image of $\gamma$ by
charts and patching restrictions thereof appropriately along finitely many
pieces of $\gamma$. Uniqueness can be proved by an open-closed argument on
$[0,1]$.

If the path is a loop ($\gamma(0)=\gamma(1)=M$) then $s_{1}\circ s_{0}^{-1}$
coincides with some ${\mathbb{C}}$-affine self map
$\phi:{\mathbb{C}}\to{\mathbb{C}}$ in a neighbourhood of
$s_{0}(M)\in{\mathbb{C}}$. The map $\phi$ and the point $s_{0}(M)$ are
independent of the choice of the representatives $s_{0}$ and $s_{1}$ of
$\tilde{s}_{0}$ and $\tilde{s}_{1}$ of the germ. The _monodromy_ of $\gamma$
is defined as the conjugacy class of $\phi$ in the group of non-constant
affine self-maps of ${\mathbb{C}}$. This class depends only on the
free777Without basepoint. homotopy class of $\gamma$. By abuse of language, we
will say that the monodromy is $z\mapsto z$ or $z\mapsto z+1$ in the
translation case and $z\mapsto\lambda z$ in the other cases. The _monodromy
factor_ is defined as $1$ in the translation case and $\lambda$ in the other
cases.

An equivalent point of view, which we will not develop here, is to endow the
universal cover $U\cal S$ of $\cal S$ based on $M$ with a similarity atlas
compatible with that of $\cal S$, and to extend the initial germ $s_{0}$ to a
global affine map $U\cal S\to{\mathbb{C}}$.

##  2\. Polygons

Gluing bounded convex polygons

Assume we are given a finite collection $(P_{j})_{j\in J}$ of bounded convex
polygons. Each polygon must have finitely many sides (aka. edges) and their
vertices are the end of these sides. As subsets of ${\mathbb{C}}$, we will
choose that polygons include their sides, i.e. they are closed for the
topology of ${\mathbb{C}}$. We allow flat angles ($\pi$ radians) at a vertex,
so keep in mind that the data of the polygon as a subset of ${\mathbb{C}}$
does not necessarily characterize the set of its vertices. Each edge is
naturally oriented by following the polygon’s boundary anticlockwise. See
Figures 1 and 2.

$P_{1}$$a$$b$$c$$P_{2}$$a$$b$$c$ Figure 1. An example of polygon gluing
pattern. The letters indicate the edge pairing. Here the quotient is
homeomorphic to a sphere and has three vertices and three edges. $a$$a$$b$$b$
Figure 2. Another example. This is the classical torus fundamental domain, but
with a random quadrilateral instead of a square. Here there is only one
polygon and only one vertex after projection to the quotient.

Consider the collection of all the sides $e$ with their polygons $P_{j}$ and
denote them $e{\,:\,}j$. Assume we are given a pairing888A partition into sets
of cardinality $2$. of them. From this, one can define a compact oriented
topological manifold $\cal S$ of dimension $2$, as follows: for each pair,
choose an ordering of the pair as $e_{1}{\,:\,}j_{1},e_{2}{\,:\,}j_{2}$; note
that the case $j_{1}=j_{2}$ is allowed by the pairing, but not the case
$e_{1}{\,:\,}j_{1}=e_{2}{\,:\,}j_{2}$; there is a unique complex affine map
$s$ sending $e_{1}$ to $e_{2}$ and reversing their orientations. Let $\cal P$
be the disjoint union of all polygons, whose elements will be denoted
$z{\,:\,}j$ to distinguish two elements of differently indexed polygons with
the same affix, which may happen. Let $\cal S$ be $\cal P$ quotiented by
$z{\,:\,}j_{1}\sim s(z){\,:\,}j_{2}$ for all the paired edges as above. Call
$\pi:\cal P\to\cal S$ the quotient map.999We use the same notation for
Archimedes’ constant, context should prevent confusion.

Interior points of polygons are not identified with any other points. An
interior point $z:j$ of an edge is identified with exactly one other point
$s(z):j^{\prime}$, which sits in the interior of the matching edge. Depending
on the situation, a vertex can be matched with any finite number of other
vertices.

Local model at vertices

Let us look at what the quotient looks like at a vertex $\pi(v_{0}:j_{0})$.
Consider all the $v:j$ such that $\pi(v:j)=\pi(v_{0}:j_{0})$: these are the
vertices of the $P_{j}$ that project to the same point. (Note that it is quite
possible that a $P_{j}$ has several vertices with the same image by $\pi$.)
Near these vertices the polygon $P_{j}$ looks like a sector with some opening
angle and the point $v$ belongs to two edges, one termed “before” and the
other one ”after” so that the sequence before$\ \rightarrow\ $sector$\
\rightarrow\ $after follows the anticlockwise order. Consider the data
$(v,e,j)$, that we will call _flag_ , where $e$ is the edge that is termed
“before”. Denote $e^{\prime}$ the edge termed “after”: it is glued via some
affine map $s$ to some edge of some other polygon $P_{j^{\prime}}$ and we call
_successor_ of $(v,e,j)$ the flag $(s(v),s(e^{\prime}),j^{\prime})$. Following
the consecutive gluings, we get from flag to flag and eventually back to the
initial flag since there are finitely many polygons. By explicit
transformations, we can map homeomorphically each sector to sectors of the
same opening with apex of affix $0$ and so that the gluings, except maybe one,
become the identity. This shows that the quotient is indeed a manifold at the
vertices.

Removing vertices and getting a similarity surface

Let $\cal V\subset\cal S$ denote the set of all vertices (after passing to the
quotient). On $\cal S^{\prime}=\cal S-\cal V$ one can define an atlas of a
similarity surface as follows: inside each polygon
$\pi(\operatorname{Int}P_{j})$, take $\pi^{-1}$ as a chart. Near an edge
$e=\pi(e_{1}{\,:\,}j_{1})=\pi(e_{2}{\,:\,}j_{2})$ with endpoints removed (call
this $\operatorname{Int}e$), we consider two cases. If $P_{1}\neq P_{2}$ then
take the neighbourhood
$V=\pi(\operatorname{Int}P_{1})\cup\operatorname{Int}e\cup\pi(\operatorname{Int}P_{2})$
and the chart $\phi:V\to
s(\operatorname{Int}P_{1})\cup\operatorname{Int}e_{2}\cup\operatorname{Int}P_{2}$
mapping $\pi(z:P_{1})$ to $s(z)$ and $\pi(z:P_{2})$ to $z$ (this coincides in
$\operatorname{Int}e$). If $P_{1}=P_{2}$ we do the same but using only non-
overlapping neighbourhoods of $\operatorname{Int}e_{1}$ and
$\operatorname{Int}e_{2}$ in $P_{1}$ instead of the full polygon. In both
cases, the sets $s(\operatorname{Int}P_{1})$ and $\operatorname{Int}P_{2}$
cannot overlap because we took convex polygons and $s$ reverses the
orientation of edges. See Figure 3.

Figure 3. Continuation of Figure 1; by gluing the polygons along their paired
edges with appropriate ${\mathbb{C}}$-affine maps, we get local chart of a
similarity surface in neighbourhoods of the edges minus their endpoints.

Conformal erasability of the singularities

$a$$a$$b$$b$$p$$p$$\tau/6$$\tau/12$ Assume the vertex $p$ is shared by these
two triangles and no other. $b$$a$$b$ Here we glued the side pair labelled $a$
together and added a exp-polar grid. We ignore what the dashed sides are
attached to. $b$$b$$u$ Image by $z\mapsto\log z$. Blue vector has affix
$u=-\log 2+i\tau/4$ and gluing of sides $b$ becomes translation by $u$.
$i\tau$ Applying $z\mapsto\frac{i\tau}{u}z+\operatorname{const}$ turns this
set (the small copy) into a bigger one and the vector $u$ into $i\tau$. $a$$b$
Last we apply the exponential, which performs the gluing of the side pair $b$,
and gives us a Riemann chart near the vertex $p$. Figure 4. Riemann chart near
a vertex for a specific example. We have $\lambda=1/2$ and $\theta=\tau/4$,
the monodromy is $z\mapsto\frac{i}{2}z$. (For convenience we used $\tau=2\pi$
in this figure.)

Seeing $\cal S^{\prime}$ as a Riemann surface,101010Or a finite union thereof,
see LABEL:foo:bar. the vertices could be either _punctures_ (the Riemann
surface can be extended by adding the point and an appropriate chart) or
_holes_ (no such extension can be done; this happens iff the vertex has a
neighbourhood in $\cal S^{\prime}$ conformally isomorphic to a ring of finite
modulus: $1<|z|<1+\varepsilon$). Let us prove that we are in the first case by
completing the Riemann surface $\cal S^{\prime}$ at the vertices, thus
promoting $\cal S$ to a Riemann surface. Let $p\in\cal S$ be a vertex.
Consider the circularly ordered sequence of flags $(v,e,j)$ associated to $p$
as described earlier in this section, and their associated angular sectors.
Let $\theta>0$ denote the sum of the angle of the sectors. When we pass from a
flag $(v,e,j)$ to its successor $(s(v),s(e^{\prime}),j^{\prime})$, there is an
affine map $s(z)=az+b$ performing the transition. Let $\lambda$ denote the
_inverse_ of the product of the dilation factors $|a|$. This product is _not_
necessarily $1$. The monodromy111111Defined at the end of
LABEL:sec:simsurfdef. of the similarity surface for an anticlockwise loop
winding once around the vertex is equal to

$z\mapsto{\lambda}{e^{i\theta}}z.$

######  Remark.

It is easy to get confused and believe that the monodromy would be the inverse
of this function. If we scale an rotate the different sectors to attach them
successively, we get a big sector (living in the universal cover of
${\mathbb{C}}^{*}$ if $\theta>2\pi$) of which there remains only one pair of
sides to glue together by some similitude. Call them “before” and after so
that what we successively meet in the anticlockwise order is “before”, sector,
“after”. Then the monodromy is the similitude that sends “before” to “after”,
not the other way round.

Figure 4 illustrates an example and also illustrates the construction of a
Riemann chart that we now explain. Map each sector to a horizontal band,
infinite on the left, by a branch of $z\mapsto\log(z-v)$. The gluing map from
a band to its successor is then a translation. Let us move them around in
${\mathbb{C}}$ and stack them according to these gluings. We obtain a band $B$
of height $\theta$, on which the last gluing goes from the top line to the
bottom line and is the translation by the complex vector
$u=\log(\lambda)+i\theta$. Now, for a well-chosen complex number $\alpha$, the
similarity $z\mapsto\alpha z$ send this vector to $2\pi i$ and the band to a
band that is not necessarily horizontal any more, yet is still infinite on the
left, and whose two sides are glued by the translation by $2\pi i$. A
uniformizing map for this quotient is just the exponential map $\exp$ from
$\alpha B$ to a neighborhood $U$ of $0$ in ${\mathbb{C}}^{*}$. (If we would
have sent $u$ to $-2\pi i$ instead of $2\pi i$, the band would be infinite
towards the right, and its exponential would be a neighbourhood of infinity.)
Adding $\\{0\\}$ to $U$ we get an analytic chart near the vertex $p$. We have
turned $\cal S$ into a compact Riemann surface.121212Or a finite union
thereof, see LABEL:foo:bar.

Unbounded polygons

$c$$f$$a$$a$ Figure 5. Example of an unbounded polygon with the two unbounded
sides glued together by a similitude whose center $c$ is different from the
focus $f$ of the unbounded sector.

We will need to include unbounded polygons $P$ into the game. As above, we
start by including only convex ones. They are assumed closed as subsets of
$\hat{{\mathbb{C}}}$, i.e. they include the vertex $\infty$. We call _finite_
vertex a vertex that is in ${\mathbb{C}}$, i.e. different from $\infty$. We
only consider unbounded polygons that have exactly two edges reaching
infinity, we call them _unbounded_ edges, and require that these edges are
half-lines, i.e. that each reach to a finite vertex, possibly the same. Our
requirement rules out in particular the case where an edge is a whole straight
line, but still allows the two unbounded edges to have a union that is a whole
straight line (iff. they end at the same finite vertex and make there an angle
of $\pi$).

This assumption implies that there are at least two edges, so rules out the
case $P={\mathbb{C}}$, which has no edge, and also rules out a half plane with
no finite vertex. It also rules out the case where the complement of $P$ would
have more than one components. Such a polygon would be bounded by two parallel
lines and there would either be at least an edge from infinity to itself or
four edges reaching infinity. However, we can recover all these cases by
pasting appropriate polygons satisfying our assumptions (see also the last
paragraph of the present section).

With these conditions each unbounded polygon has only one infinite vertex and
each unbounded edge is a half line. We now consider the previous
constructions, but allowing the presence of unbounded polygons that satisfy
the conditions. We have to assume that the pairing between edges satisfies
that unbounded edges are paired with unbounded edges. Moreover, the
orientation reversing ${\mathbb{C}}$-affine map sending an unbounded
$e_{1}:j_{1}$ to an unbounded $e_{2}:j_{2}$ exists but is not anymore unique.
So the data that must be given prior the construction has to include the
choice of this map for each such pair (for a visual way to do this, one can
mark unbounded edges with supplementary non-vertex points, that must match
under the gluings, as in Section 4.4).

Then the construction can be carried out and we still get a topological
surface $\cal S$, but the set of vertices $\cal V\subset\cal S$ now includes a
new kind of element: _infinite_ vertices. These are the vertices which come
from the vertex at infinity of an unbounded polygon.

Again something new may happen: conformally, _infinite vertices are not
anymore necessarily punctures_. An example is given in Section 4.4. To the
infinite vertex of an unbounded convex polygon satisfying our restrictions,
one can associate the angle between the two edges reaching to infinity. It is
a non-negative number. In the example of Section 4.4 this angle is $0$.

To a infinite vertex $v$ of the quotient let us again associate the sum of the
angles at infinity of the unbounded polygons whose infinite vertex projects to
$v$. Then a sufficient condition for an infinite vertex to be a puncture is
that this sum is positive. The uniformizing map is obtained in a similar way
as for finite vertices, with a few complications. We will provide indications
but leave details to the reader.

An infinite (closed) sector is of the form $a+[R,+\infty)\cdot
e^{i[\theta_{1},\theta_{2}]}$ for some $R\geq 0$ and some $a\in{\mathbb{C}}$
that we will call its _focus_. An half infinite strip is the image of the set
$[0,1]\times[0,+\infty)\subset{\mathbb{R}}^{2}\simeq{\mathbb{C}}$ by a complex
affine map. Here is a list of complications. The angle of an unbounded polygon
could be $0$. The total angle at an infinite vertex could be $0$. The focuses
of the infinite sectors do not necessarily coincide with the finite vertices
of the unbounded edges. The focuses of the infinite sectors do not necessarily
match under the gluing maps of these edges. Taking a logarithm based on a
focus would not conjugate the gluing (a similarity) to a map as simple as a
translation (and if the two finite vertices of the two unbounded edges do not
coincide, we of course cannot base the log on both).

Instead we take subsets of the sectors and strips that are neighbourhoods of
infinity and glue them in a universal cover $U{\mathbb{C}}^{*}$ of
${\mathbb{C}}^{*}$ into one big set sitting in finitely many sheets of the
cover (one sheet is enough near $\infty$ if $\theta<2\pi$). A neighbourhood of
infinity in this set is a sector or a strip. Its two sides must be glued
together according to some ${\mathbb{C}}$-affine map (the vertices are now
excluded from of the picture, except the one at infinity) whose factor is the
monodromy factor (or its inverse, depending on the which side is the domain
and which side is the range). If we order the two sides of the infinite
sector/strip in the clockwise order (this is the opposite of the ordering we
chose for finite vertices) then the monodromy sends the first one to the
second one (as in the finite case). The monodromy factor must be of the form

$\lambda e^{-i\theta}$

for some $\lambda>0$. Now there are several cases

  1. (1)

$\theta>0$.

    1. (a)

Either the affine map $s$ has a unique fixed point. Then we can take a
logarithm based on this fixed point. The image of the two sides will be almost
horizontal curves131313Because we work close to infinity. If bothered by that,
one can make these curves actual horizontals by shifting the original edges,
which can, in a neighbourhood of infinity, be achieved by cutting and pasting
pieces of the sectors. that extend infinitely to the right and that we must
glue by a translation of vector $v=\log\lambda-i\theta$. Multiplication by
$2\pi i/v$ followed by $\exp$ realizes this gluing.

    2. (b)

Or the affine map $s$ is the identity. Then the total angle is a multiple of
$2\pi$, and the two curves to be glued together have the same projection by
$\phi:U{\mathbb{C}}^{*}\to{\mathbb{C}}^{*}$. The quotient is a $k$-fold cover
of a punctured neighborhood of infinity. A uniformization is given by a
determination of $z\mapsto z^{-1/k}$.

    3. (c)

Or the affine map $s$ is a translation by a non-zero vector $a$. In this case
$\lambda=1$, and $\theta$ must be a multiple of $2\pi$:

$\theta=k2\pi.$

By composing everything with $z\mapsto a^{-1}z$ we can assume that $a=1$. By
cutting and pasting we can assume that the edges to glue have their infinite
direction of argument $0$. We provide explicitly a formula _whose inverse_
realizes the gluing:

$\phi(z)=z^{-k}+\frac{1}{2\pi i}\log z$

near infinity, where one takes the branch of $\log z$ whose imaginary part
lies in $[0,2\pi]$. We omit the details.

  2. (2)

$\theta=0$.

    1. (a)

Either the gluing map is a translation, which can be taken as $z\mapsto z+2\pi
i$ by an appropriate ${\mathbb{C}}$-affine change of coordinate. Then a
uniformization is provided by $z\mapsto\exp(z)$.

    2. (b)

Or the gluing map is of the form $z\mapsto az+b$ for some $a\in{\mathbb{R}}$
with $a>0$ and $a\neq 0$. Then the vertex at infinity is not a puncture but a
hole.

Signed angle and monodromy factor

To unify the formulas for the monodromy we define the _signed angle_
$\sigma\in{\mathbb{R}}$ of a vertex as $\sigma=\theta$ if the vertex is finite
and $\sigma=-\theta$ if the vertex is infinite, where $\theta\geq 0$ is the
total angle defined in the previous paragraphs. Then the factor of the
monodromy of a small loop winding anticlockwise around the singularity is in
both cases of the form

$\lambda e^{i\sigma}$

for some $\lambda>0$. Note that the monodromy factor determines $\lambda$
completely but only determines $\sigma$ modulo $2\pi i$.

Even fancier polygons

In fact we can use polygons that are non-convex, or not simply connected, and
without restrictions on the type of unbounded edges or their number, still
finite. Indeed such polygons can be cut further, possibly introducing
auxiliary finite vertices (in which case, obviously, the resulting similarity
surface structure will have an erasable singularity at these vertices), so as
to respect the previous restrictions and yield the same objects $\cal S$ and
$\cal S^{\prime}$ once we give back to the latter the auxiliary vertices. Then
one thing will have to be kept in mind: that if a polygon
$P\subset{\mathbb{C}}$ has $n$ unbounded complementary components in
${\mathbb{C}}$, then it must be considered as having $n$ distinct infinite
vertices (before the gluing; after, this number may get reduced).

One could also use polygons that overlap themselves, i.e. are spread over
${\mathbb{C}}$ in an non-injective way, but this section is already long
enough.

##  3\. The Schwarz-Christoffel formula

###  3.1. Christoffel symbol

Consider any similarity surface $\cal S^{\prime}$. Recall that it is in
particular a Riemann surface. Assume that we are given a similarity chart
$s:U\to{\mathbb{C}}$ and a Riemann chart $r:U\to{\mathbb{C}}$ on the
same141414We can always restrict a chart to an open subset of its domain and
add it to the atlas, this gives the same structure on $\cal S^{\prime}$. open
subset $U\subset\cal S^{\prime}$. Consider the map $\phi=s\circ r^{-1}$: it
expresses a similarity chart in a Riemann chart. Note that $\phi$ is
holomorphic, by definition of a Riemann surface atlas. Now if we are given two
similarity charts $s_{1}$ and $s_{2}$ on $U$ and if $U$ is connected then by
definition of a similarity atlas, there must exist $a\in{\mathbb{C}}^{*}$ and
$b\in{\mathbb{C}}$ such that $s_{2}=a\,s_{1}+b$. (In fact the definition
ensures the existence of $a$ and $b$ locally, and analytic continuation
ensures they are constant on the connected set $U$.) It follows that

$\phi_{2}=a\,\phi_{1}+b$

hence

$\phi_{2}^{\prime}=a\,\phi_{1}^{\prime}$

thus

$\frac{\phi^{\prime\prime}_{1}}{\phi^{\prime}_{1}}=\frac{\phi^{\prime\prime}_{2}}{\phi^{\prime}_{2}}.$

For any holomorphic function on an open subset of ${\mathbb{C}}$ we now use
the notation

$N\phi=N(\phi):=\frac{\phi^{\prime\prime}}{\phi^{\prime}}.$

Assume we are given a subset $U$ of $\cal S^{\prime}$. Note that for every
point $M\in U$ there are similarity charts defined in a neighbourhood of $M$
but there may fail to exist a similarity chart that would be defined on the
entirety of $U$, even if there is a Riemann chart $r$ defined on $U$.

If we are given a Riemann chart $r:U\subset\cal S^{\prime}\to{\mathbb{C}}$, we
can use it to endow $r(U)$ with the similarity atlas $\\{\phi=s\circ
r^{-1}\\}$ where $s$ varies in the similarity atlas of $\cal S^{\prime}$. Note
that if we denote $V$ the domain of $s$ then the domain of $\phi$ is $r(U\cap
V)$.

We will thus momentarily study open subsets $O$ of ${\mathbb{C}}$ with a
similarity surface atlas whose charts $\phi$ are holomorphic. By the above
computation, all similarity charts have the same quantity $N(\phi)$ at a given
$z\in O$, so even if there may fail to be a single chart $\phi$ on $O$, there
is a well-defined holomorphic function $\zeta:O\to{\mathbb{C}}$ such that for
all similarity charts $s:V\subset O\to{\mathbb{C}}$, then

$N\phi=\zeta$

on $V$. It is easy to recover $\phi$ from $\zeta$, as the equation
$\phi^{\prime\prime}/\phi^{\prime}=\zeta$ is equivalent to
$\log\phi^{\prime}=\int\zeta$ (locally), so

$\phi=\int\exp\left(\int\zeta\right)$

holds locally. The integration constants make $\phi$ known locally only up to
post-composition by ${\mathbb{C}}$-affine maps, which is coherent with the
fact that we have a similarity atlas:
$\phi=b+\int\exp\left(c+\int\zeta\right)=b+a\int\exp\left(\int\zeta\right)$
with $a=e^{c}$. Note also that, for a loop $\gamma$ contained in a single
chart $O$ the monodromy factor of a loop will be equal to the exponential of
the integral of $\zeta$ along this loop:

$\lambda=\exp\left(\int_{\gamma}\zeta(z)dz\right).$

Let us come back to abstract similarity surfaces $\cal S^{\prime}$. Up to now
we only expressed similarity charts in _one_ Riemann chart, but what happens
if we change the Riemann chart? Assume $r_{1}$ and $r_{2}$ are both defined on
$U\subset\cal S^{\prime}$ and let $\psi=r_{1}\circ r_{2}^{-1}$. Then the
expressions $\phi^{(i)}=s\circ r_{i}^{-1}$ of a similarity chart $s$ via the
maps $r_{i}$ are related by $\phi^{(2)}=\phi^{(1)}\circ\psi$. It is easy, for
holomorphic functions, to compute $N(\phi\circ\psi)$:

(1) $N(\phi\circ\psi)=\psi^{\prime}\times N(\phi)\circ\psi+N(\psi).$

For our situation let us write this as

(2)
$\zeta^{(2)}=\psi^{\prime}\times\zeta^{(1)}\circ\psi+\frac{\psi^{\prime\prime}}{\psi^{\prime}}.$

where $\zeta^{(i)}$ is the function of the previous paragraph, for the
similarity atlas $\phi^{(i)}$ on the subset $r_{i}(U)$ of ${\mathbb{C}}$.

######  Remark.

Equation 1 is reminiscent of the formula of a change of variable for a
$1$-form. In fact $N\phi$ does not express a $1$-form on the Riemann surface
$\cal S^{\prime}$ but a _Christoffel symbol_ in complex dimension one.
Christoffel symbols are the coefficients that appear in the infinitesimal
expression of affine connections in charts, see LABEL:sub:affcoconf. Equation
1 is also reminiscent of a similar formula for the _Schwarzian derivative_
$S\phi$. In fact if we look at surfaces with atlases whose change of charts
are homographies instead of affine maps, the operator $S$ will play the role
of the operator $N$.

If we have an isolated point in ${\mathbb{C}}-U$, then by eq. 2 the polar part
of $\zeta$ behaves under a change of variable exactly like the polar part of a
holomorphic $1$-form. In particular the residue “$\operatorname{res}$” is an
invariant. This is coherent with the fact that the monodromy of the similarity
surface around this singularity is independent of any choice of Riemann
surface charts and is a conjugacy class of affine map whose linear factor is
equal to $\exp(2\pi i\operatorname{res})$.

###  3.2. Polygon gluings and residue of the Christoffel symbol near puncture
type singularities

We place ourselves here in the situation of Section 2 where a topological
surface $\cal S$ and a similarity surface $\cal S^{\prime}=\cal S-\cal V$ were
constructed by gluing bounded or unbounded polygons, where $\cal V$ is the set
of vertices, and we make the assumption that all infinite vertices have an
angle sum $\theta$ that is non-zero. We saw that in this case that the atlas
associated to $\cal S^{\prime}$ extends to a Riemann surface atlas of $\cal
S$. Let us study more closely the Christoffel symbol near the vertices.

For any finite vertex we constructed a Riemann chart whose image is a
neighborhood of $0$ and in which the similarity charts are branches of
$z\mapsto z^{\alpha}$ for the complex number
$\alpha=(\log\lambda+i\theta)/2\pi i$ where $\theta$ is the sum of the angles
of the polygons at this vertex and $\lambda$ is such that the monodromy factor
at the vertex is $\lambda e^{i\theta}$. For any branch of $z\mapsto
z^{\alpha}$, a direct computation gives

$Nz^{\alpha}=\frac{\alpha-1}{z}.$

It follows (see eq. 2 and the consequence on the polar part explained after
it) that in any other Riemann chart, the Christoffel symbol $\zeta$ will have
a simple pole at the vertex, of residue

(3) $\operatorname{res}=\alpha-1=\frac{\log(\lambda)+i\theta}{2\pi i}-1.$

In particular

${\operatorname{Re}\,}\operatorname{res}>-1.$

A similar construction was done for infinite vertices for which $\theta>0$.
The conclusion was similar: the monodromy has factor $\lambda e^{-i\theta}$
for some $\lambda>0$ and there is a Riemann chart mapping a neighbourhood of
this vertex to a neighbourhood of $0$ and for which similarity charts are
given by branches of $z\mapsto z^{\alpha}$ with
$\alpha=(\log\lambda-i\theta)/2\pi i$ or, if the monodromy is a non-identity
translation (hence $\lambda=1$ and $\theta=k2\pi$), by branches of $z\mapsto
z^{-k}+\frac{1}{2\pi i}\log z$. The Christoffel symbol has in both cases a
_simple pole_ at the vertex, of residue

(4) $\operatorname{res}=\alpha-1=\frac{\log(\lambda)-i\theta}{2\pi i}-1$

in this chart and thus in any other Riemann chart. In particular

${\operatorname{Re}\,}\operatorname{res}<-1.$

######  Remark.

The notion of signed angle $\sigma$ in Section 2 allows to unify Equations 3
and 4: in both cases

$\operatorname{res}=\frac{\log(\lambda)+i\sigma}{2\pi i}-1.$

The monodromy factor $\lambda e^{i\sigma}$ has a unified expression in terms
of $\operatorname{res}$:

$\lambda e^{i\sigma}=\exp(2\pi i\operatorname{res}).$

###  3.3. Polygons gluing to a sphere and the Schwarz-Christoffel formula

We now make the supplementary assumption that the topological surface $\cal S$
is homeomorphic to a sphere. (We still place ourselves in the situation of
Section 3.2, which we recall: $\cal S$ is obtained by gluing together finitely
many bounded or unbounded polygons; all infinite vertices are moreover
required to have an angle sum $\theta$ that is non-zero, so that all
singularities are punctures and the Riemann surface atlas of $\cal S^{\prime}$
extends to $\cal S$.) Then by the _Poincaré Koebe_ theorem, $\cal S$ is
isomorphic as a Riemann surface to the Riemann sphere $\hat{\mathbb{C}}$. Let
$F:\cal S\to\hat{\mathbb{C}}$ be such an isomorphism. Let
$\\{z_{1},\ldots,z_{m}\\}\subset\hat{\mathbb{C}}$ be the image by $F$ of the
vertices. The point $\infty\in\hat{\mathbb{C}}$ may or may not be one of them.
For each vertex $v_{k}$, let $\theta_{k}$ be the total angle,
$\sigma_{k}=\pm\theta_{k}$ the signed angle, and $\lambda_{k}$ the dilation
ratio of this vertex. Note that $F$ gives us a _global chart_ of the Riemann
surface $\cal S-F^{-1}(\infty)$. Removing vertices, $F$ is a global Riemann
chart for $\cal S^{\prime}-F^{-1}(\infty)$. The associated Christoffel symbol
$\zeta$ is a holomorphic function from ${\mathbb{C}}-\\{z_{1},\ldots,z_{m}\\}$
to ${\mathbb{C}}$.

Let $\operatorname{res_{k}}$ denote the residue of $\zeta$ at the vertex
$z_{k}$.

######  Theorem 1.

We have

$\zeta(z)=\sum_{\stackrel{{\scriptstyle
k=1}}{{z_{k}\neq\infty}}}^{m}\frac{\operatorname{res}_{k}}{z-z_{k}}.$

Moreover

$\sum_{k=1}^{m}\operatorname{res}_{k}=-2.$

###### Proof.

By the above discussion, $\zeta$ has at most a simple pole at each vertex
$z_{k}\neq\infty$ and we know the corresponding residue
$\operatorname{res}_{k}$. To analyse what happens at infinity we use the
change of variable $w=\psi(z)=1/z$, call $\tilde{\zeta}$ the expression of the
Christoffel symbol in the coordinate $w$ and use Equation 2:

$\zeta(z)=-z^{-2}\tilde{\zeta}(z^{-1})-2z^{-1}.$

If $\infty$ is not a vertex then $\zeta(z)=-2z^{-1}+\cal O(z^{-2})$ as
$z\to\infty$. If $\infty$ is a vertex $z_{k}$ then
$\tilde{\zeta}(w)=\frac{\operatorname{res}_{k}}{w}+\cal O(1)$ as $w\to 0$,
from which:

$\zeta(z)=(-\operatorname{res}_{k}-2)z^{-1}+\cal O(z^{-2})$

as $z\to\infty$. In all cases: $\zeta$ tends $0$ to at $\infty$.

Let $f(z)$ denote the sum of $\operatorname{res}_{k}/(z-z_{k})$ as in the
statement of the theorem. The difference $\zeta-f$ is a holomorphic function
on ${\mathbb{C}}$ minus finitely many points, whose singularities are erasable
and which tends to $0$ at infinity. It follows that $\zeta-f=0$. This proves
the first claim. Moreover, from the above it follows that
$\zeta(z)-f(z)=\left(-2-\sum\operatorname{res}_{k}\right)z^{-1}+\cal
O(z^{-2})$ as $z\to\infty$, whence the second claim. ∎

######  Remark.

That the sum of residues is $-2$ (in the case of $\hat{\mathbb{C}}$) and not
$0$ is another difference between the Christoffel symbol and $1$-forms. It can
be proved more generally for all Christoffel symbol on $\hat{\mathbb{C}}$ with
finitely many singularities (polar or essential) by using Equation 2 as above
and expressing $\int\zeta(z)dz$ on a big circle in two ways, using the
residues on each side. Another (equivalent) way is to use that the difference
of two Christoffel symbols is a $1$-form and to use that the sum of residues
of a $1$-form is always $0$ to reduce the computation to a particular symbol
(for instance $\zeta=0$). Last, let us remark that this formula generalizes to
genus $g$ compact Riemann surfaces: the sum of the residues is then equal to
$2g-2$.

From Theorem 1, the similarity charts $\phi$ satisfy locally that

$\phi=a+b\int\prod_{\stackrel{{\scriptstyle
k=1}}{{z_{k}\neq\infty}}}^{m}\left(z-z_{k}\right)^{\operatorname{res}_{k}}.$

######  Remark.

This is a generalization of the _Schwarz-Christoffel formula_. See [Ché08] for
more details.

###  3.4. Example

$a$$b$$a$$b$$A$$B$$C$$C$ Figure 6. The example of Section 3.4. Left: the edges
with the same label are glued together. Right: image under a uniformization.

In Figure 6 we illustrate the following example: one polygon is an equilateral
triangle and the other is a isocele rectangle triangle. The three sides of the
first are glued to the three sides of the second as on the figure on the left,
so that the quotient is homeomorphic to the sphere. It has three vertices $A$,
$B$ and $C$, three edges and of course two polygons. We get the following,
where $\theta$ is the total angle and $\lambda$ the monodromy factor (using
$\tau=2\pi$ on the left to clarify the computations):

$\displaystyle A:$ $\displaystyle\theta$
$\displaystyle=\frac{\tau}{6}+\frac{\tau}{8}=\frac{7}{24}\tau$
$\displaystyle\lambda$ $\displaystyle=\sqrt{2}$
$\displaystyle\operatorname{res}$
$\displaystyle=\frac{7}{24}-1+\frac{\log\sqrt{2}}{2\pi i}$ $\displaystyle B:$
$\displaystyle\theta$
$\displaystyle=\frac{\tau}{6}+\frac{\tau}{8}=\frac{7}{24}\tau$
$\displaystyle\lambda$ $\displaystyle=1/\sqrt{2}$
$\displaystyle\operatorname{res}$
$\displaystyle=\frac{7}{24}-1-\frac{\log\sqrt{2}}{2\pi i}$ $\displaystyle C:$
$\displaystyle\theta$
$\displaystyle=\frac{\tau}{6}+\frac{\tau}{4}=\frac{5}{12}\tau$
$\displaystyle\lambda$ $\displaystyle=1$ $\displaystyle\operatorname{res}$
$\displaystyle=\frac{5}{12}-1$

In the global chart $\hat{\mathbb{C}}$ where $C$ is at $\infty$, $A$ at $-1$
and $B$ at $1$ we get

$\zeta=\frac{\operatorname{res}_{A}}{z+1}+\frac{\operatorname{res}_{B}}{z-1}.$

On Figure 6 we see on the left the two triangles with two different hatchings
(think of them as a kind of coordinate systems); on the right the
corresponding Riemann surface $\cal S^{0}$ has been mapped to
$\hat{\mathbb{C}}$ as above and we drew the image of the hatchings using the
methods of [Ché08], Section 4.3.2 to follow the geodesics associated to
$\zeta$.

###  3.5. Any Christoffel symbol

Given an open subset $U$ of ${\mathbb{C}}$ and a holomorphic function
$\zeta:U\to{\mathbb{C}}$, we can define an atlas of similarity surface whose
Christoffel symbol is $\zeta$, by solving locally $N\phi=\zeta$. Assume that
$\zeta$ has an isolated singularity $z_{0}\in{\mathbb{C}}$ which is a simple
pole of residue $\operatorname{res}$.

######  Lemma 1.

Under the conditions above, if $\operatorname{res}\notin\\{-1,-2,\ldots\\}$
then there is a holomorphic change of coordinate $z\mapsto w$ sending
$z_{0}\to 0$ such that similarity charts are expressed as branches of
$w\mapsto w^{\operatorname{res}+1}$, up to affine maps, and hence the
Christoffel symbol takes the expression

$\tilde{\zeta}(w)=\frac{\operatorname{res}}{w}.$

###### Proof.

By a translation we can assume that $z_{0}=0$. We have the power series
expansion as $z\to 0$:

$\zeta(z)=\frac{\operatorname{res}}{z}+a_{0}+a_{1}z+a_{2}z^{2}+\ldots$

from which one antiderivative satisfies

$\int\zeta=\operatorname{res}\times\log z+o(1)$

hence

$\exp\left(\int\zeta\right)=z^{\operatorname{res}}\exp(o(1))=\sum_{n=0}^{+\infty}b_{n}z^{\operatorname{res}+n}$

for some sequence $b_{n}\in{\mathbb{C}}$ with $b_{0}=1$. Since
$z^{\operatorname{res}+n}$ has several values, we need to specify a meaning:
we temporarily work in the universal cover of ${\mathbb{C}}^{*}$, let $\log z$
be a global branch and define
$z^{\operatorname{res}+n}=\exp((\operatorname{res}+n)\log
z)=\exp(\operatorname{res}\times\log z)z^{n}$. This generalized power series
expansion has complex exponents but positive radius of convergence and we can
integrate term by term to get

$\int\exp\left(\int\zeta\right)=\sum_{n=0}^{+\infty}b_{n}\frac{z^{\operatorname{res}+n+1}}{\operatorname{res}+n+1}=z^{\operatorname{res}+1}\exp(R(z))$

for some holomorphic function $R$ of $z$ (i.e. without singularity at $0$).
Then

$w=z\exp\frac{R(z)}{\operatorname{res}+1}$

does the job. ∎

Note that we did not claim local uniqueness of the coordinate $w$ that
satisfies the conclusion of the lemma. We do not wish to study the uniqueness
problem in this article.

######  Remark.

Let us state how to deal with the remaining cases (we give them without proofs
because we will not use these normal forms). In the case where
$\operatorname{res}=-1$ we can also get $\tilde{\zeta}(w)=-1/w$, which has
branches of $w\mapsto\log w$ as similarity charts up to affine maps. In the
case $\operatorname{res}=-k\in\\{-2,-3,\ldots\\}$, we can reduce to either
$\tilde{\zeta}(w)=-k/w$ or to $\tilde{\zeta}(w)=-k/w+w^{k-2}$, and these two
cases are mutually exclusive. The first one has $w\mapsto 1/w^{1-k}$ as
similarity charts but the similarity charts of the second one has a
complicated expression. We may prefer an alternative form, for example one
which gives branches of $w\mapsto\frac{-1}{w^{k-1}}+\log w$ as similarity
charts up to affine maps, in which case
$\displaystyle\tilde{\zeta}(w)=\frac{-k}{w}+\frac{w^{k-2}}{1+\frac{w^{k-1}}{(k-1)}}$.

An isolated singularity of a Christoffel symbol with a simple pole with

${\operatorname{Re}\,}(\operatorname{res})>-1$

is called _bounded_. It is called _unbounded_ if

${\operatorname{Re}\,}(\operatorname{res})<-1.$

We do not give a particular name to the case
${\operatorname{Re}\,}(\operatorname{res})=-1$.

According to the lemma above, near any _bounded_ isolated singularity of a
similarity surface, the latter is isomorphic to a bounded sector of apex $0$
and opening $\theta=2\pi({\operatorname{Re}\,}(\operatorname{res})+1)$ (if
$\theta>2\pi$, the sector sits in the universal cover of ${\mathbb{C}}^{*}$)
endowed with the canonical atlas of ${\mathbb{C}}$, and whose two sides are
glued by the similarity $z\mapsto e^{2\pi i\operatorname{res}}z$ from $\arg
z=\theta_{0}$ to $\arg z=\theta_{0}+\theta$.

Near any _unbounded_ singularity of _non-integer_ residue, this is the same
but the sector is a neighbourhood of infinity and has opening
$\theta=-2\pi({\operatorname{Re}\,}(\operatorname{res})+1)$, and the
similarity is still $z\mapsto e^{2\pi i\operatorname{res}}z$ and maps $\arg
z=\theta_{0}$ to $\arg z=\theta_{0}-\theta$.

We leave to the reader the task to determine local models for the remaining
cases if they wish. Such models will not be used here.

A convenient change of variables for the unbounded type singularities

Assume that $0$ is a simple pole of $\zeta$ of unbounded type
(${\operatorname{Re}\,}\operatorname{res}<-1$) and consider the Laurent series
expansion

$\zeta(z)=\frac{\operatorname{res}}{z}+a_{0}+a_{1}z+a_{2}z^{2}+\ldots.$

Close to $0$ the term $\frac{\operatorname{res}}{z}$ is dominant, so it is
natural to try and compare the affine charts $\phi$ of $\zeta$ to the affine
charts $z\mapsto z^{1+\operatorname{res}}$ of the “ideal” Christoffel symbol
$\frac{\operatorname{res}}{z}$. For this we make the change of variable

$\tilde{z}=z^{1+\operatorname{res}}$

and express $\zeta$ in the $u$ coordinate. Note that the correspondence
$z\longleftrightarrow\tilde{u}$ is a bijection (and usually only well defined)
from the universal cover $U{\mathbb{C}}^{*}$ of ${\mathbb{C}}^{*}$ to
$U{\mathbb{C}}^{*}$, via

$\log\tilde{z}=(1+\operatorname{res})\times\log z$

where $\log:U{\mathbb{C}}^{*}\to{\mathbb{C}}$ is a well-defined bijection.

Tending to $0$

If $\operatorname{res}\in{\mathbb{R}}$ then $|z|\longrightarrow 0$ $\iff$
$|\tilde{z}|\longrightarrow+\infty$. But if
$\operatorname{res}\notin{\mathbb{R}}$ then the set $|z|<r$, where $r>0$,
correspond to a set of $\tilde{z}\in U{\mathbb{C}}^{*}$ whose boundary spirals
inward to $0$ as
$({\operatorname{Im}\,}\operatorname{res})\times(\arg\tilde{z})\to-\infty$ and
outward to $\infty$ as
$({\operatorname{Im}\,}\operatorname{res})\times(\arg\tilde{z})\to+\infty$:
indeed the set of values of $\log\tilde{z}$ is the half plane

(5) $HU(r):=\big{\\{}{\tilde{z}\in
U{\mathbb{C}}^{*}}\,\big{|}\,{{\operatorname{Re}\,}\frac{\log\tilde{z}}{1+\operatorname{res}}<\log
r}\big{\\}},$

it is delimited by a straight line that is not vertical and this half plane
contains a neighbourhood of $+\infty$ in ${\mathbb{R}}$, because
${\operatorname{Re}\,}(1+\operatorname{res})<0$. In any cases, we have
$|z|\longrightarrow 0$ iff $\tilde{z}$ enters and stays in every $HU(r)$.

$\tilde{z}$-sectors

Since $\tilde{z}\in U{\mathbb{C}}^{*}$, there is a well defined function
$\arg\tilde{z}$ taking values in ${\mathbb{R}}$. The sector in $\tilde{z}$
coordinates defined by $\arg\tilde{z}\in(\alpha_{0},\alpha_{1})$ corresponds
in $\log u$ coordinate to the strip
${\operatorname{Im}\,}\log\tilde{z}\in(\alpha_{0},\alpha_{1})$. In this strip,
the subset for which the corresponding $z$ belongs to $B(0,r)$ is the
intersection with $\log HU(r)$. In $\tilde{z}$ coordinate, this intersection
is a subset of the sector with a spiralling neighbourhood of $0$ removed, see
Figure 7. We call this set

$U(\alpha_{0},\alpha_{1},r)=\big{\\{}{\tilde{z}\in
U{\mathbb{C}}^{*}}\,\big{|}\,{\alpha_{0}<\arg u<\alpha_{1},\
|z|<r}\big{\\}}\subset U{\mathbb{C}}^{*}$

where $\arg\tilde{z}={\operatorname{Im}\,}\log\tilde{z}$ is well-defined on
$U{\mathbb{C}}^{*}$ and $\tilde{z}\mapsto z$ also. For any given
$U=U(\alpha_{0},\alpha_{1},r)$ we have

$\log|z|={\operatorname{Re}\,}\left(\frac{1}{1+\operatorname{res}}\right)\log|\tilde{z}|+\cal
O(1)$

where $\cal O(1)$ is quantity bounded over all $U$ with a bound that depends
continuously on $(\alpha_{0},\alpha_{1},r)$ for $\alpha_{0}<\alpha_{1}$ and
$0<r$. In particular, under the constraint $\tilde{z}\in
U(\alpha_{0},\alpha_{1},r)$, we have $|z|\longrightarrow
0\iff|\tilde{z}|\longrightarrow+\infty$. Finally, note that $U$ contains the
sector defined by $\arg\tilde{z}\in(\alpha_{0},\alpha_{1})$ and
$|\tilde{z}|>R$ where $R=R(\alpha_{0},\alpha_{1},r,\operatorname{res})$ can be
computed easily.

$0$$0$$B(0,r)$$\times(1+\operatorname{res})$$\exp$$\exp$$z^{(1+\operatorname{res})}$$U(\alpha_{0},\alpha_{1},r)$$\log
HU(r)$ Figure 7. Example of domain
$U(\alpha_{0},\alpha_{1},r)=\big{\\{}{\tilde{z}}\,\big{|}\,{\arg\tilde{z}\in[-\alpha_{0},\alpha_{1}]\text{
and }|z|<r}\big{\\}}$ in the case
${\operatorname{Re}\,}\operatorname{res}<-1$. Dots correspond to $\arg
z\equiv\pi\bmod 2\pi$, dashes to $\arg\tilde{z}=\alpha_{0}$ or $\alpha_{1}$.

######  Lemma 2.

Assume that ${\operatorname{Re}\,}\operatorname{res}<-1$. There is some $r>0$
that depends on $\zeta$ and such that the following holds. Let $\phi$ be any
solution151515They exists since $B(0,r)-\\{0\\}$ lifts to a simply connected
subset of $U{\mathbb{C}}^{*}$ on the part of $U{\mathbb{C}}^{*}$ on which
$|z|<r$, of $\phi^{\prime\prime}(z)/\phi^{\prime}(z)=\zeta(z)$. The quantity
$\phi$ can be interchangeably considered as depending on $z$ or $\log z$ or
$\tilde{z}$, etc. Then there exists $a\in{\mathbb{C}}^{*}$ such that:

  1. (1)

there exists a holomorphic function $z\mapsto r_{1}(z)$ on $B(0,r)$ such that

$\frac{\partial\phi}{\partial\log\tilde{z}}=a\times\tilde{z}\times(1+r_{1}(z))$

and $r_{1}(0)=0$;

  2. (2)

for all $\alpha_{0},\alpha_{1}\in{\mathbb{R}}$ with $\alpha_{0}<\alpha_{1}$,
there exists $C,C^{\prime}\in{\mathbb{R}}$ and a holomorphic function
$r_{2}:U(\alpha_{0},\alpha_{1},r)\to{\mathbb{C}}$ such that we have,
$\forall\tilde{z}\in U(\alpha_{0},\alpha_{1},r)$,

$\phi=a\tilde{z}+r_{2}(\tilde{z})$

with

$|r_{2}(\tilde{z})|\leq\max\left(|z|\\!\times\\!|\tilde{z}|,1\right)(C^{\prime}+C\log|\tilde{z}|).$

In particular

$\phi\underset{|\tilde{z}|\to\infty}{\sim}a\tilde{z}.$

###### Proof.

We restrict $\zeta$ to some $B(0,r)$ so that

$\zeta(z)=\frac{\operatorname{res}}{z}+h(z)$

with $h$ bounded, say $|h|\leq M$. Let $v=\log z$ and let us express the
Christoffel symbol $\hat{\zeta}$ in the $v$ coordinate: since $z=e^{v}$,
formula (2) gives

$\hat{\zeta}(v)=e^{v}\zeta(e^{v})+1$

and using the development of $\zeta$ above we get

$\hat{\zeta}(v)=1+\operatorname{res}+e^{v}h(e^{v}).$

Let $\hat{\phi}(v)=\phi(z)$. Then $\hat{\phi}$ is a solution of
$\hat{\phi}^{\prime\prime}/\hat{\phi}^{\prime}=\hat{\zeta}$. Integrating on
the half plane ${\operatorname{Re}\,}v<\log r$ we get that

$\log(\hat{\phi}^{\prime}(v))=c_{0}+(1+\operatorname{res})v+g(e^{v})$

where $g$ is the antiderivative of $h$ mapping $0$ to $0$ and
$c_{0}\in{\mathbb{C}}$ is a constant. Hence

$\frac{\partial\phi}{\partial\log\tilde{z}}=\frac{1}{1+\operatorname{res}}\hat{\phi}^{\prime}(v)=a\tilde{z}(1+r_{1}(z))$

for $a=e^{c_{0}}\in{\mathbb{C}}^{*}$ and for the holomorphic bounded function
$r_{1}:B(0,r)\to{\mathbb{C}}$ mapping $0$ to $0$ and defined by
$r_{1}(z)=\exp(g(z))-1$. This proves the first point.

We have $|r_{1}(z)|\leq c|z|$ for some $c>0$ and thus

(6) $\left|\frac{\partial\phi}{\partial\log\tilde{z}}-a\tilde{z}\right|\leq
c^{\prime}|\tilde{z}|\times|z|$

with $c^{\prime}=c|a|$. Let us abbreviate $U=U(\alpha_{0},\alpha_{1},r)$. The
set of $\tilde{z}\in\partial U$ such that $|z|=r$ is a compact set for which
$\phi$ is hence bounded. Any element $\tilde{z}\in U$ satisfies
$\log\tilde{z}=t+\log\tilde{z}_{0}$ for a $t>0$ and a $\tilde{z}_{0}$ as
above. We have that $|\tilde{z}|$ and $e^{t}$ are comparable in the sense that
their quotient is bounded away from $0$ and $\infty$ by factors depending only
on $U$. Similarly $|z|$ and $e^{-\alpha t}$ are comparable where

$\alpha=-{\operatorname{Re}\,}(1/(1+\operatorname{res}))>0.$

Consider the path $t\mapsto\tilde{z}(t)=e^{t}\tilde{z}_{0}$ and the
corresponding path $\log\tilde{z}(t)=\log\tilde{z}_{0}+t$. Let us integrate
eq. 6 along the latter path. Note that an antiderivative of $\tilde{z}$ with
respect to $\log\tilde{z}$ is $\tilde{z}$. We have
$|\tilde{z}(t)|=|u_{0}|\int_{0}^{t}e^{t}$ and $|z(t)|=|v_{0}|e^{-\alpha t}$.
Hence

$\left|\phi(z(t))-a\tilde{z}(t)-\phi(z(0))+a\tilde{z}_{0}\right|\leq
c^{\prime}|\tilde{z}_{0}||z_{0}|\int_{0}^{t}e^{s}e^{-\alpha s}ds$

If $\alpha\geq 1$ then $e^{s}e^{-\alpha s}\leq 1$ whence
$\int_{0}^{t}e^{s}e^{-\alpha s}ds\leq t$. If $\alpha\leq 1$ then
$e^{s}e^{-\alpha s}\leq e^{(1-\alpha)t}$ whence $\int_{0}^{t}e^{s}e^{-\alpha
s}ds\leq t\leq te^{(1-\alpha)t}$. ∎

The sector in $\tilde{z}$ coordinates defined by
$\arg\tilde{z}\in(\alpha_{0},\alpha_{1})$ corresponds in $\log\tilde{z}$
coordinate to the strip
${\operatorname{Im}\,}\log\tilde{z}\in(\alpha_{0},\alpha_{1})$. In this strip,
the subset for which the corresponding $z$ belongs to $B(0,r)$ is the
intersection with $\log HU(r)$. In $\tilde{z}$ coordinate, this intersection
is a subset of the sector with a spiralling neighbourhood of $0$ removed, see
Figure 7. We call this set

$U(\alpha_{0},\alpha_{1},r)=\big{\\{}{\tilde{z}\in
U{\mathbb{C}}^{*}}\,\big{|}\,{\alpha_{0}<\arg\tilde{z}<\alpha_{1},\
|z|<r}\big{\\}}\subset U{\mathbb{C}}^{*}$

where $\arg\tilde{z}={\operatorname{Im}\,}\log\tilde{z}$ is well-defined on
$U{\mathbb{C}}^{*}$ and $\tilde{z}\mapsto z$ also. Note that it contains the
sector defined by $\arg\tilde{z}\in(\alpha_{0},\alpha_{1})$ and
$|\tilde{z}|>R$ where $R=R(\alpha_{0},\alpha_{1},r,\operatorname{res})$ can be
computed easily.

######  Corollary 3.

In the conditions of Lemma 2, for any
$\alpha_{0}<\alpha_{0}^{\prime}<\alpha_{1}^{\prime}<\alpha_{1}$ there exists
$0<r^{\prime\prime}<r^{\prime}\leq r$ such that for any
$\theta\in{\mathbb{R}}$, if one denotes $S$ the (simply connected) set of
$z\in U{\mathbb{C}}^{*}$ for which $\tilde{z}\in
U(\alpha_{0},\alpha_{1},r^{\prime})$, then the restriction of $\phi$ to $S$
avoids $0$, and has a lift to $U{\mathbb{C}}^{*}$ that is analytic, injective
and whose image contains $a\times
U(\alpha_{0}^{\prime},\alpha_{1}^{\prime},r^{\prime\prime})$, where $a$ is the
factor of Lemma 2.

###### Proof.

It follows from the definitions and from the estimate of the second point in
Lemma 2 using, for instance, the argument principle on $\phi$ if
$\alpha_{1}-\alpha_{0}<2\pi$ or for the general case the argument principle on
$\log\phi$.161616To avoid the $\log$ one can instead invoke homological
arguments. ∎

For the next lemma, we call bounded resp. unbounded sector of apex $0$ subsets
of ${\mathbb{C}}$ the form

$\big{\\{}{z\in{\mathbb{C}}^{*}}\,\big{|}\,{\alpha_{0}<\arg z<\alpha_{1}\text{
and }|z|<r}\big{\\}}$

resp.

$\big{\\{}{z\in{\mathbb{C}}^{*}}\,\big{|}\,{\alpha_{0}<\arg z<\alpha_{1}\text{
and }|z|>r}\big{\\}}$

with $\alpha_{1}-\alpha_{0}\in(0,2\pi]$ and $r>0$.

######  Lemma 4.

For any geodesic parametrized by $t$ and tending to a bounded resp. unbounded
singularity $z_{0}$:

  1. (1)

The geodesic takes a finite time to reach the singularity in the bounded case,
and an infinite time in the unbounded case.

  2. (2)

There exists $r>0$ and a simply connected subset $U$ of
$B(z_{0},r)-\\{z_{0}\\}$, and an (injective) affine chart
$\phi:U\to{\mathbb{C}}$ whose image is a bounded resp. unbounded sector with
apex $0$, and such that $\phi$ sends the geodesic, for $t$ large enough, into
the bisecting line of the sector, tending to $0$ for a bounded singularity and
to $\infty$ for an unbounded one.

Note that the hypotheses exclude the case
${\operatorname{Re}\,}\operatorname{res}\neq-1$.

###### Proof.

We distinguish two overlapping cases:
$\operatorname{res}\notin\\{-2,-3,\ldots\\}$ and
${\operatorname{Re}\,}\operatorname{res}<-1$.

In the case $\operatorname{res}\notin\\{-2,-3,\ldots\\}$ the lemma follows
from the local model deduced from Lemma 1, both in the bounded and unbounded
case. In the bounded case, this model is the quotient of a sector (with the
canonical atlas of ${\mathbb{C}}$) by a similarity fixing its apex, and a
geodesic is a locally straight line in the atlas, hence one tending to the
singularity must end up being a ray through the apex, as can be seen by the
same trick as the fact that a billiard trajectory in a sector can only undergo
finitely many bounces: if we have a a straight line geodesic that is not aimed
at the apex then patching finitely many copies of the sector by the similarity
covers a sector of angle $>\pi$ that will contain the geodesic until it
escapes. In the unbounded case, tending to $0$ in the $z$ coordinate
corresponds to tending to infinity in the model, hence the time tends to
infinity, and similarly to the previous case, one see that the geodesic
eventually remains in one fundamental domain, then we apply a translation to
the chart so that the geodesic radiates from $0$, then the fundamental domain
still contains an unbounded sector with apex $0$.

In the case ${\operatorname{Re}\,}\operatorname{res}<-1$ we first focus on
proving that the time tends to infinity. For this we recall the first point of
the conclusion of Lemma 2:

$\frac{\partial\phi}{\partial\log\tilde{z}}=a\times\tilde{z}\times(1+r_{1}(z))$

where $r$ is holomorphic and bounded in $z\in B(0,r)$ and $r_{1}(0)=0$. By
taking $r$ smaller we can assume that $r^{\prime}(z)$ is bounded too. We have

$\arg\frac{\partial\phi}{\partial\log\tilde{z}}=\arg
a+{\operatorname{Im}\,}\log\tilde{z}+{\operatorname{Im}\,}\log(1+r_{1}(z)).$

By taking $r$ small enough we can assume that this quantity has modulus
$<1/10$ for all $\tilde{z}\in HR(r)$. We have

$\frac{\partial\,{\operatorname{Im}\,}\log(1+r_{1}(z))}{\partial\log\tilde{z}}={\operatorname{Im}\,}\left(\frac{r^{\prime}_{1}(z)}{1+r_{1}(z)}\times\frac{z}{1+\operatorname{res}}\right).$

By taking $r$ small enough we can assume that this quantity also has modulus
$<1/10$ for all $\tilde{z}\in HR(r)$. We let $t_{0}$ for which for all $t\geq
t_{0}$ then $|z(t)|<r$ and we now only consider times $t\geq t_{0}$. As $t$
varies and $z$ varies accordingly along the geodesic, the variables
$\tilde{z}$, and $\phi$ can be considered as depending on $t$. The geodesic
has the property that $\frac{\partial\phi}{\partial t}$ remains constant, in
particular the argument of this quantity remains constant. Since
$\frac{\partial\log\tilde{z}}{\partial t}=\frac{\partial\phi}{\partial
t}\left/\frac{\partial\phi}{\partial\log\tilde{z}}\right.$ we have that, as a
function of $t$, the quantity $\log\tilde{z}$ follows a field line of a vector
field whose direction is given by

$\theta(\log\tilde{z})=c-\,{\operatorname{Im}\,}\log\tilde{z}-{\operatorname{Im}\,}\log(1+r_{1}(z))$

for some constant $c\in{\mathbb{R}}$ that depends on $a$ and on the direction
of the geodesic in the $\phi$ coordinate. Isoclines of this vector field are
given by the level lines of $\theta\bmod 2\pi$. By the estimates on
$\log(1+r_{1}(z))$ and its derivative with respect to $\log\tilde{z}$ we have
that:

  * $-$

the isocline of angle $\beta$ is a union of curves $I_{\beta+2k\pi}$,
$k\in{\mathbb{Z}}$, each contained within distance $<1/10$ of the horizontal
line of ordinate $c-\beta$;

  * $-$

the tangent to $I_{\beta}$ at every point makes an angle
$<\operatorname{asin}\frac{1}{10}$ with the horizontal direction;

  * $-$

each of these curves extends from $\partial HU(r)$ to infinity;

  * $-$

each is asymptotic at infinity to the horizontal curve of imaginary part
$c-\beta$.

We have that $\theta(\log\tilde{z})$ strictly decreases when $\log\tilde{z}$
moves upwards along a vertical line; we will say that $\log\tilde{z}$ is
_above_ $I_{\beta}$ if $\theta(\log\tilde{z})<\beta$ and _below_ if
$\theta(\log\tilde{z})>\beta$ (note the inversion). We have
$\operatorname{asin}\frac{1}{10}<\pi/4$ which, together with the above points,
implies the following: Consider the isocline $I_{\beta}$ with
$\beta=2k\pi-\pi/4$. If $\log\tilde{z}(t_{0})$ is below $I_{\beta}$ then for
$t>t_{0}$ the curve $\log\tilde{z}(t)$ is disjoint $I(\beta)$ so
$\log\tilde{z}(t)$ stays below $I_{\beta}$. Similarly if
$\log\tilde{z}(t_{0})$ is above $I_{\beta}$ with $\beta=2k\pi-\pi/4$ then
after $t_{0}$, $\log\tilde{z}$ stays above $I_{\beta}$. Let $\beta_{*}$ denote
the isocline on which the geodesic starts. We take the smallest $\beta_{0}\in
2\pi{\mathbb{Z}}-\pi/4$ such that $\beta_{0}\leq\beta_{*}$ and the biggest
$\beta_{1}\in 2\pi{\mathbb{Z}}+\pi/4$ such that $\beta_{1}\geq\beta_{*}$ then
$\log\tilde{z}(t)$ has to remain between the isoclines of index $\beta_{0}$
and $\beta_{1}$. Note that the value of $\beta_{1}-\beta_{0}$ is either
$2\frac{\pi}{4}$ or $2\pi+2\frac{\pi}{4}$, depending on $\alpha_{*}$. In
particular, $\log\tilde{z}(t)$ stays within a horizontal strip of height $\leq
2\pi+2\frac{\pi}{4}+\frac{2}{10}$. Now the only way for $z(t)$ to tend to $0$
while $\log\tilde{z}(t)$ has bounded imaginary part is that
${\operatorname{Re}\,}\log\tilde{z}(t)\longrightarrow+\infty$. By the second
point of Lemma 2, we have that $\phi(t)$ tends to infinity, hence the geodesic
is defined for an infinite amount of time. This proves the first claim.

With similar arguments the estimates on $r_{1}(z)$ for $z$ small imply that
$\log\tilde{z}(t)$ is asymptotic to a horizontal line of imaginary part
$\alpha_{\infty}=c-2k\pi$ for some $k\in{\mathbb{Z}}$. Let
$\alpha_{0}=c-\beta_{1}-1/10$ and $\alpha_{1}=c-\beta_{0}+1/10$. Then
$\alpha_{0}<\alpha_{\infty}<\alpha_{1}$. We choose any $\alpha^{\prime}_{0}$
and $\alpha^{\prime}_{1}$ such that
$\alpha_{0}<\alpha^{\prime}_{0}<\alpha_{\infty}<\alpha^{\prime}_{1}<\alpha_{1}$
and apply Corollary 3: the set $\tilde{z}\in U(\alpha_{0},\alpha_{1},r)$
corresponds bijectively to a set of $\phi\in U{\mathbb{C}}^{*}$ that contains
$a\times U(\alpha^{\prime}_{0},\alpha^{\prime}_{1},r^{\prime\prime})$. (If one
would need a definite margin between $\alpha_{\infty}$ and
$\alpha_{0}^{\prime}$ and $\alpha_{1}^{\prime}$, it is always possible to
change the value of $\alpha_{0}$ to a smaller value and $\alpha_{1}$ to a
bigger value.) We saw that $\tilde{z}(t)$ remains in the domain
$U(\alpha_{0},\alpha_{1},r^{\prime})$ and since $\phi(t)\sim a\tilde{z}(t)$,
$\phi(t)$ eventually enters and remains in the domain $a\times
U(\alpha_{0}^{\prime},\alpha_{1}^{\prime},r^{\prime\prime})$. We have
$\phi(t)=a^{\prime}t+b$ for some $a^{\prime}\in{\mathbb{C}}^{*}$ and
$b\in{\mathbb{C}}$ and
$U(\alpha_{0}^{\prime},\alpha_{1}^{\prime},r^{\prime\prime})$ contains a
sector with apex $0$ and central direction $a^{\prime}$. We can replace $\phi$
by $\phi-b$ so that $\phi(t)=a^{\prime}t$ is now a radial line. Then the image
of this $\phi$ still contains an unbounded sector with apex $0$ and central
direction $a^{\prime}$, possibly with smaller angle. ∎

###  3.6. Resting place and paths to a bounded singularity

As explained in LABEL:sec:simsurfdef, given a continuous path
$\gamma:[0,1]\to\cal S^{\prime}$ and a germ of chart $[\phi_{0}]$ (the square
brackets refer to the fact that the germ is an equivalence class) at the
beginning of the path $\gamma(0)$, there is a way to follow this germ along
the path $\gamma(t)$ as $[\phi_{t}]$. The value $\phi_{t}(\gamma(t))$ is well-
defined and the function $\tilde{\gamma}:t\mapsto\phi_{t}(\gamma(t))$ is
called the _development_ of the path $\gamma$ w.r.t. $[\phi_{0}]$. Let us call
the point $\phi_{1}(\gamma_{1})$ the _resting place_ of the development
(_principal value_ could be a nice alternative denomination).

Can this be done for a path tending to a bounded singularity? The germ will in
general degenerate, but the development will have a limit, provided we take
some precaution. To see this, let us work in a Riemann chart where the
singularity is at $0$ and where for some $\alpha\in{\mathbb{C}}$ with
${\operatorname{Re}\,}\alpha>0$, the similarity charts include the branches of
$z\mapsto z^{\alpha}$. Then if ${\operatorname{Im}\,}\alpha\neq 0$ and $z$
winds along a circle around $0$ many times in the sense opposite to the sign
of ${\operatorname{Im}\,}\alpha$, then $z^{\alpha}$ tends to infinity. Or if
$z$ spirals down to $0$ too slowly, winding this way, then $z^{\alpha}$ still
tends to $\infty$. On the other hand, if $z$ tends to $0$ along a well-defined
tangent direction, then $z^{\alpha}$ tends to $0$, and this limit $0$ does not
depend on the direction. Hence the development $\tilde{\gamma}$ of the path
$\gamma$ may or may not converge, but if we replace its last moments by a
straight line (in any given Riemann chart), its development has a limit that
is independent of the replacement done, and we can thus associate to $\gamma$
and $[\phi_{0}]$ a well-defined resting place.

Consider two paths $[0,1]\to\cal S$ that both start from the same point and
end at the same bounded singularity. Assume that they take their values in
$\cal S^{\prime}$ except at $t=1$ and assume that they are homotopic in $\cal
S$ by a homotopy rel. $\\{0,1\\}$ (i.e. that fixes the ends) and that takes
value in $\cal S^{\prime}$ between the ends. Then their developments starting
from the same germ $[\phi_{0}]$ have the same resting place (indeed the
development can be followed under the homotopy and the resting place stays
immobile).

##  4\. Holomorphic dependence

###  4.1. Holomorphic dependence of special coordinates.

We now state and prove a holomorphic dependence statement related to Lemma 1.
We will restrict to parameter spaces of dimension one to simplify the
presentation and since in the application in LABEL:part:beltrami we only need
that.

Let $\zeta_{0}$ be a particular map satisfying the conditions of Lemma 1. We
assume that $r>0$ and that we are given complex numbers
$\operatorname{res}_{t}$ and $z_{t}$ that depend holomorphically on $t\in
B(0,r)$. We assume that $\varepsilon>0$, that $z_{t}\in B(z_{0},\varepsilon)$
and that we have a family $(t,z)\mapsto\zeta_{t}(z)$ for $t\in B(0,r)$ and
$z\in B(z_{0},\varepsilon)-\\{z_{t}\\}$ given by

$\zeta_{t}(z)=\frac{\operatorname{res}_{t}}{z-z_{t}}+h_{t}(z)$

where the map $(t,z)\in B(0,r)\times B(z_{0},\varepsilon)\mapsto
h_{t}(z)\in{\mathbb{C}}$ is analytic.

######  Lemma 5.

Under these conditions, for $\eta$ small enough, there exists $r^{\prime}>0$
such that for $t\in B(0,r^{\prime})$, one can choose the change of variable
$(z\mapsto w)$ of Lemma 1 to be defined on $B(z_{0},\eta)$ and depend
holomorphically on $t$.

###### Proof.

In Remark Remark. ∎

If for $t=0$ we have $\operatorname{res}_{0}\in\\{-2,-3,\ldots\\}$ then
$\zeta_{0}$ is not in the situation of Lemma 1. Instead we adapt Corollary 3.
Recall the definition of the $\tilde{z}$-coordinate given before Corollary 3:
$\tilde{z}=z^{1+\operatorname{res}}$, which is well-defined if $z$ and
$\tilde{z}$ are considered as elements of the universal cover
$U{\mathbb{C}}^{*}$ of ${\mathbb{C}}^{*}$. Recall also the definition of the
sets $U(\alpha_{0},\alpha_{1},r)=\big{\\{}{\tilde{z}\in
U{\mathbb{C}}^{*}}\,\big{|}\,{\alpha_{0}<\arg\tilde{z}<\alpha_{1},\
|z|<r}\big{\\}}$. Complete $U{\mathbb{C}}^{*}$ into a topological space
$\widehat{U}{\mathbb{C}}^{*}=U{\mathbb{C}}^{*}\cup\\{\hat{0}\\}$ by adding a
point $\hat{0}$ whose neighbourhoods consists in the sets containing
$V_{\varepsilon}=\\{\hat{0}\\}\cup$ the set of points whose projection to
${\mathbb{C}}^{*}$ has modulus $<\varepsilon$.

######  Lemma 6.

Assume that ${\operatorname{Re}\,}\operatorname{res}<-1$. Denote
$\tilde{z}=z^{1+\operatorname{res}_{t}}\in U{\mathbb{C}}^{*}$ where $z\in
U{\mathbb{C}}^{*}$. Then there exists $\varepsilon>0$ and
$0<r^{\prime\prime}<r^{\prime}<r$ such that for any $\theta\in{\mathbb{R}}$
and any $t\in B(0,\varepsilon)$, there exists a simply connected open subset
$S_{t}\subset U{\mathbb{C}}^{*}$, $a_{t}\in{\mathbb{C}}^{*}$ and a holomorphic
function $\phi_{t}:S_{t}\to{\mathbb{C}}$ such that

  * $-$

$z_{t}+\pi(S_{t})\subset B(z_{0},r)$ where $\pi$ is the projection
$U{\mathbb{C}}^{*}\to{\mathbb{C}}^{*}$;

  * $-$

$S_{t}$ depends continuously on $t$ by an isotopy of
$\widehat{U}{\mathbb{C}}^{*}$;

  * $-$

in $\tilde{z}$ coordinate, the image of $S_{t}$ contains
$U(\theta-\pi/2,\theta+\pi/2,r^{\prime})$;

  * $-$

$\phi_{t}^{\prime\prime}(z)/\phi_{t}^{\prime}(z)=\zeta_{t}(z_{t}+\pi(z))$;

  * $-$

$\phi_{t}$ depends holomorphically on $t$;

  * $-$

$a_{t}$ depends holomorphically on $t$;

  * $-$

for each fixed $t$, $\phi_{t}(z)\sim a_{t}\tilde{z}$ when $z\longrightarrow 0$
within $S_{t}$;

  * $-$

the image of $\phi_{t}$ contains $a_{t}\times
U(\theta-\pi/4,\theta+\pi/4,r^{\prime\prime})$.

###### Proof.

By the change of variable $z^{\prime}=z-z_{t}$, possibly reducing $r$, it is
enough to treat the case where $z_{t}=0$ for all $t$. Recall that
$\tilde{z}\mapsto z$ is well-defined and note that $z$ depends analytically on
$(\tilde{z},\operatorname{res})$. Let $\tilde{\zeta}_{t}$ be the pull-back of
$\zeta_{t}$ in $\tilde{z}$-coordinate: it depends analytically on
$(t,\tilde{z})$ and hence there exists a global solution
$\phi=\tilde{\phi}_{t}$ of
$\phi^{\prime\prime}/\phi^{\prime}=\tilde{\zeta}_{t}$ on $HU(r)\subset
U{\mathbb{C}}^{*}$ that depends holomorphically on $t$. Consider any.

The variable $\log\tilde{z}$ depends analytically on $t$ and by looking at the
proof of Lemma 2, one sees that $r_{1}(z)$ and thus $a$ also depend
analytically on $t$. Moreover, in Lemma 2, all estimates can be taken uniform
provided $t$ is small enough and $\alpha_{0}$ and $\alpha_{1}$ are fixed. The
same holds for Corollary 3. ∎

###  4.2. Holomorphic dependence: resting place

Let $U$ be an open subset of ${\mathbb{C}}$. Consider two distinct points
$z_{0}\neq z_{1}$. Consider the set $\cal G(z_{0},z_{1})$ of paths
$\gamma:[0,1]\to U$ from $z_{0}$ to $z_{1}$ such that $\gamma(s)\neq z_{1}$
when $s\neq 1$. Consider a homotopy class $[\gamma]$ in the set above, i.e. a
path-connected component of this set. To state the next lemma we endow the set
$\Gamma$ of all $(z_{0},z_{1},[\gamma])$ with a complex manifold structure.
For this we need to determine when two homotopy classes are close when their
endpoints do not match, i.e. we need a topology on $\Gamma$.

Given two paths $\gamma$, $\tilde{\gamma}$ in ${\mathbb{C}}$, let

$d(\gamma,\tilde{\gamma})=\sup_{s\in[0,1]}|\tilde{\gamma}(s)-\gamma(s)|.$

Let $\cal G$ be the set of paths $\gamma$ in $U$ that satisfy $\forall
s\in[0,1]$, $\gamma(s)\neq\gamma(1)$: it is the disjoint union of the $\cal
G(z_{0},z_{1})$ for $z_{0}\neq z_{1}$, both in $U$. For all $\varepsilon>0$
and all $\gamma\in\cal G$, let $V_{\gamma,\varepsilon}$ be the set of
$(\tilde{z}_{0},\tilde{z}_{1},[\tilde{\gamma}])$ where $\tilde{\gamma}\in\cal
G$, $d(\gamma,\tilde{\gamma})<\varepsilon$, $\tilde{z}_{0}=\tilde{\gamma}(0)$
and $\tilde{z}_{1}=\tilde{\gamma}(1)$.

We will use the following facts, see Section 4.4 for proofs.

  * $-$

There exists171717It is easily seen to be unique, since a set is open iff. it
is a neighbourhood of all its points. a topology on $\Gamma$ such that for all
$\gamma\in\cal G$, the collection of $V(\gamma,\varepsilon)$ forms a basis of
neighbourhoods.

  * $-$

For this topology the sets $V(\gamma,\varepsilon)$ are open.

  * $-$

It is Hausdorff separated.

  * $-$

The map

$(z_{0},z_{1},[\gamma])\in\Gamma\mapsto(z_{0},z_{1})\in{\mathbb{C}}^{2}$

is a local homeomorphism.

This allows to endow the set $\Gamma$ of all $(z_{0},z_{1},[\gamma])$ with a
two dimensional complex manifold structure, locally given by the position of
the points $z_{0}$ and $z_{1}$.

######  Lemma 7.

Let $U$ be an open subset of ${\mathbb{C}}$ (or of a Riemann surface).
Consider a moving point $z_{0}(t)$ and a family of Christoffel symbols $t\in
B(0,r)\mapsto\zeta_{t}$ of the form

$\zeta_{t}(z)=h_{t}(z)+\frac{\operatorname{res}_{t}}{z-z_{1}(t)}$

with $z_{0}(t)$, $z_{1}(t)$ and $\operatorname{res}_{t}$ holomorphic in $t$,
$h_{t}(z)$ holomorphic in $z\in U$ and $t$, and for all $t$: $z_{0}(t)\neq
z_{1}(t)$ and ${\operatorname{Re}\,}\operatorname{res}_{t}>-1$ (i.e. the
singularity $z_{1}(t)$ of $\zeta_{t}$ is bounded, possibly removable). Choose
$a_{t}\in{\mathbb{C}}^{*}$ and $b_{t}\in{\mathbb{C}}$ holomorphic in $t$.
Consider the unique germ of similarity chart $\phi_{t}$ of $\zeta_{t}$ with
$\phi_{t}(z_{0}(t))=b_{t}$ and $\phi_{t}^{\prime}(z_{0}(t))=a_{t}$. Let
$X_{t}:=(z_{0}(t),z_{1}(t),[\gamma_{t}])\in\Gamma$ vary continuously (hence
holomorphically) with $t$. To all this data, associate the resting place
$c_{t}$ (see Section 3.6) of the development of $\phi_{t}$ along $\gamma_{t}$.
Then $c_{t}$ depends holomorphically on $t$.

###### Proof.

See Remark Remark. ∎

###  4.3. Holomorphic dependence: following a saddle connection when varying
$\zeta$

A notion of _geodesic_ can be defined on similarity surfaces: as parametrized
curves which in similarity charts have a constant speed vector, a condition
that is indeed independent of the chosen similarity chart. Geodesics do not
need to be injective. A geodesic from a singularity to another one is called a
_saddle connection_ by analogy with dynamical sytems.

A geodesic reaching a bounded singularity does it in finite time: see Lemma 4.
A saddle connection between bounded singularities is hence parametrized by a
bounded interval. We can reparametrize it by the segment $(0,1)$ if needed, by
a real affine change of variable.

######  Proposition 8.

Let $U$ be an open subset of ${\mathbb{C}}$ and $z^{*}_{0}$, $z^{*}_{1}$ two
distinct points in $U$. Assume $\zeta^{*}$ is a Christoffel symbol holomorphic
on $U-\\{z^{*}_{0},z^{*}_{1}\\}$ and that

$\zeta^{*}(z)=h^{*}(z)+\frac{\operatorname{res}^{*}_{0}}{z-z^{*}_{0}}+\frac{\operatorname{res}^{*}_{1}}{z-z^{*}_{1}}$

with $h$ holomorphic and bounded on $U$. Assume that
${\operatorname{Re}\,}\operatorname{res}^{*}_{k}>-1$ for $k\in\\{0,1\\}$ (i.e.
the singularities $z_{k}$ are either bounded or better: removable). Last,
assume that a saddle connection $f^{*}:(0,1)\to U$ exists between the two
singularities. Then there exists $\varepsilon>0$ such that for all Christoffel
symbols $\zeta$ of the form

$\zeta(z)=h(z)+\frac{\operatorname{res}_{0}}{z-z_{0}}+\frac{\operatorname{res}_{1}}{z-z_{1}}$

with $h:U\to{\mathbb{C}}$ holomorphic, $\sup_{z\in
U}|h(z)-h^{*}(z)|<\varepsilon$, and for all $k\in\\{0,1\\}$,
$|z_{k}-z_{k}^{*}|<\varepsilon$ and
$|\operatorname{res}_{k}-\operatorname{res}_{k}^{*}|<\varepsilon$, then there
exists a saddle connection $f:(0,1)\to U-\\{z_{0},z_{1}\\}$ from $z_{0}$ to
$z_{1}$, and if $h$, $\operatorname{res}_{k}$, $z_{k}$ depend analytically on
a complex parameter $\tau$ then $f$ depends analytically on $\tau$.

###### Proof.

In Remark Remark. ∎

We extend the notion of saddle connection to include geodesics from a bounded
singularity to an unbounded one (or in the opposite direction), under the
assumption that the unbounded singularity has positive total angle. Such a
geodesic can be parametrized by $t\in(0,+\infty)$, but there is no more
uniqueness of the parametrization: it can be reparametrized by a linear change
of the variable $t$. Moreover, since the total angle of the infinite vertex is
positive, it follows that one can vary in some extent the direction, in a
similarity chart, along which $\infty$ is reached. There are at least two ways
to recover uniqueness: either by specifying the initial speed at the bounded
singularity. Or by specifying a marked point which it has to go through, and
which must be the image of $t=1$ by the parametrization. We choose the second
approach here and in Section 4.4.

######  Proposition 9.

Let $U$ be an open subset of ${\mathbb{C}}$ and $z^{*}_{0}$, $z^{*}_{1}$,
$z^{*}_{2}$ three distinct points in $U$. Assume $\zeta^{*}$ is a Christoffel
symbol holomorphic on $U-\\{z^{*}_{0},z^{*}_{2}\\}$ and that

$\zeta^{*}(z)=h^{*}(z)+\frac{\operatorname{res}^{*}_{0}}{z-z^{*}_{0}}+\frac{\operatorname{res}^{*}_{2}}{z-z^{*}_{2}}$

with $h$ holomorphic and bounded on $U$. Assume that
${\operatorname{Re}\,}\operatorname{res}^{*}_{0}>-1$ and
${\operatorname{Re}\,}\operatorname{res}^{*}_{2}<-1$ (i.e. $z_{0}$ is a
bounded singularity and $z_{2}$ is an unbounded one). Last, assume that a
saddle connection $f^{*}:(0,+\infty)\to U$ exists from $z^{*}_{0}$ to
$z^{*}_{2}$ with $f^{*}(1)=z^{*}_{1}$. Then there exists $\varepsilon>0$ such
that for all Christoffel symbols $\zeta$ of the form

$\zeta^{0}(z)=h(z)+\frac{\operatorname{res}_{0}}{z-z_{0}}+\frac{\operatorname{res}_{1}}{z-z_{2}}$

with $h:U\to{\mathbb{C}}$ holomorphic, $\sup_{z\in
U}|h(z)-h^{*}(z)|<\varepsilon$, for all $k\in\\{0,1,2\\}$,
$|z_{k}-z_{k}^{*}|<\varepsilon$ and for all $k\in\\{0,2\\}$,
$|\operatorname{res}_{k}-\operatorname{res}_{k}^{*}|<\varepsilon$, then there
exists a saddle connection $f:(0,+\infty)\to U-\\{z_{0},z_{2}\\}$ from $z_{0}$
to $z_{2}$, with $f(1)=z_{1}$, and if $h$, $\operatorname{res}_{k}$, $z_{k}$
depend analytically on a complex parameter $\tau$ then $f$ depends
analytically on $\tau$.

###### Proof.

See Remark Remark. ∎

###  4.4. Holomorphic dependence with respect to the polygons

Let us come back to our example of similarity surface $\cal S^{\prime}$
constructed by gluing polygons, and its completion into a Riemann surface
$\cal S$ that is isomorphic to the Riemann sphere via

$F:\cal S\to\hat{\mathbb{C}}.$

The map $F$ is unique up to an automorphism of $\hat{\mathbb{C}}$, i.e. a
homography. We recall that the group of homographies is sharply
$3$-transitive.

In this section we allow unbounded polygons but we assume that their angle at
infinity is always $>0$, which rules out for instance vertices with residue of
real part $-1$.

We can always subdivide the polygons further (adding vertices is allowed) to
meet the following conditions:

  1. (1)

All polygons are strictly convex,

  2. (2)

every unbounded polygon has exactly two unbounded edges, and their angle is
$>0$,

  3. (3)

$\cal S$ has at least three vertices.

The space of bounded strictly convex polygons with $n$ indexed vertices is
endowed with a complex structure, which is simply given by the affix of its
vertices. For (strictly convex) _unbounded polygons_ satisfying (2), we need
to add a supplementary information: a marked point in the interior of the
unbounded edges, but this marked point is not considered as a vertex. For any
paired unbounded sides, there is a unique orientation reversing affine map
sending one to the other and matching their marked points. Reciprocally, for
any orientation reversing affine map sending one to the other we can choose
(non unique) pairs of matching marked points.

######  Remark.

The non-injectivity of this representation, due to this non-uniqueness, is not
seen as a problem. It can be recovered by replacing the marked points by the
dilation factor $a\in{\mathbb{C}}^{*}$ of the affine map $z\mapsto az+b$
pairing the unbounded sides. Note that, anyway, the space of polygons can be
factored further by considering polygons up to affine transformations.

In the two statements below, we fix the type of each polygon $P_{j}$—number of
edges, boundedness thereof—but not the position of their vertices or marked
points, and we fix the combinatorial data of which edges are paired together.

######  Lemma 10.

Under these conditions, the residues $\operatorname{res}_{k}$ depend
holomorphically on the bounded polygons and marked unbounded polygons.

###### Proof.

Recall Equations 3 and 4 in their unified version:

$\operatorname{res}=\frac{\log\lambda+i\sigma}{2\pi i}-1$

for some $\sigma\in{\mathbb{R}}$, and that the monodromy factor of a small
loop winding anticlockwise around the vertex is $\lambda e^{i\sigma}$.

For a finite vertex, $\sigma=\theta$ is the total angle of the vertex. The
monodromy is also a product of quotients
$\displaystyle\frac{a^{\prime}-a}{b^{\prime}-b}$ where $a$, $a^{\prime}$, $b$,
$b^{\prime}$ are vertices.

For an infinite vertex $\sigma=-\theta$ and the monodromy factor is also a
product of quotients $\displaystyle\frac{a^{\prime}-a}{b^{\prime}-b}$ where
$a$, $b$ are vertices and $a^{\prime}$, $b^{\prime}$ are vertices or marked
points. The claim follows. ∎

Choose three distinct vertices $v_{1}$, $v_{2}$, $v_{3}$ and choose the unique
$F$ as above that sends them respectively to $\infty$, $0$ and $1$. Label the
other vertices up to $v_{p}$ for some $p\geq 3$. Let $z_{k}=F(v_{k})$.

######  Proposition 11.

Under these conditions and the ones stated before Lemma 10, the points $z_{k}$
depend holomorphically on the bounded polygons and marked unbounded polygons.

###### Proof.

In Section 4.4. ∎

We will also use the following complement. To state it, denote $\cal P$ the
finite collection of the polygons that we are gluing together. Let us note
that the polygons, their edges and vertices define after the gluing a cell
complex on $\cal S$. Under $F$, the cell complex decomposition maps to a cell
complex decomposition of $\hat{\mathbb{C}}$. Let us focus on its $1$-skeleton
$\operatorname{\mathit{Sk}}(F)$, together with the marked points. The edges in
this skeleton are geodesics of the similarity structure, and they are saddle
connections (they run from vertices to vertices, possibly through marked
points).

######  Proposition 12.

Under the same conditions, the $1$-skeleton $\operatorname{\mathit{Sk}}(F)$
moves holomorphically, and this motion extends to an isotopy matching the
$2$-cells.

###### Proof.

See Propositions 18 and 19 in Section 4.4. ∎

By this we mean that there is a family of homeomorphism $S_{\cal
P}:\operatorname{\mathit{Sk}}(\cal S^{0})\to\operatorname{\mathit{Sk}}(F)$
where $\cal P$ varies in the set of allowed polygons and marked unbounded
polygons, and where $\cal S^{0}$ is the abstract surface, and a cell complex,
associated to some arbitrary basepoint $\cal P^{0}$, such that $S_{\cal P}$
matches corresponding vertices and marked points and $\forall
x\in\operatorname{\mathit{Sk}}(\cal S^{0})$, $S_{\cal P}(x)$ depends
holomorphically on $\cal P$. Moreover if one uses a geodesic parametrization
of the edges (minus endpoints) by $(0,1)$, or by $(0,+\infty)$ sending the
marked point to $1$, then $S_{\cal P}$ preserves the parametrization. Last,
the $2$-cells cut by the skeletons also match in a continuous fashion.

######  Remark.

Propositions 8 and 9 already imply that saddle connections can be followed
holomorphically. The point of Proposition 12 is to prove that they form, as
expected, $\operatorname{\mathit{Sk}}(F)$.

Appendix A Proofs

A.1. Holomorphic dependence of special coordinates: proof of Lemma 5

We do not provide every detail here as it would be tedious to read. We assume
that we have a change of variable for $\zeta_{0}$ as in Lemma 1 on
$B(z_{0},\varepsilon)$. For $t$ small enough, $z_{t}$ remains in
$B(0,\varepsilon)$ and $\operatorname{res}_{t}$ remains in
${\mathbb{C}}-\\{-1,-2,\ldots\\}$. Let us note that a first change of variable
$z\mapsto z-z_{t}$ obviously depends holomorphically on $\zeta$, which allows
us to restrict to the case where $z_{t}$ does not change and is equal to $0$.
We go through the computations of Lemma 1 again:

$\int\zeta_{t}=\operatorname{res}_{t}\log z+\int h_{t}$

where $\int h_{t}$ is the antiderivative of $h_{t}$ that vanishes at the
origin. This map $\int h_{t}$ depends holomorphically on $t$ and $z$. Then

$\exp\left(\int\zeta_{t}\right)=z^{\operatorname{res}_{t}}\exp\left(\int
h_{t}\right).$

Denote $g_{t}=\exp\left(\int h_{t}\right)$ and let $g_{t}(z)=\sum
b_{n}(t)z^{n}$ be its power series expansion with respect to $z$. We have

$\int
z^{\operatorname{res}_{t}}g_{t}(z)dz=z^{\operatorname{res}_{t}+1}\sum_{n=0}^{+\infty}\frac{b_{n}(t)z^{n}}{\operatorname{res}_{t}+n+1}=z^{\operatorname{res}_{t}+1}\frac{f_{t}(z)}{\operatorname{res}_{t}+1}$

with

$f_{t}(z)=\sum_{n=0}^{+\infty}\frac{\operatorname{res}_{t}+1}{\operatorname{res}_{t}+n+1}b_{n}(t)z^{n}=\sum
a_{n}(t)z^{n}.$

This series has at least the radius of convergence of $g_{t}$. By using
Cauchy’s estimates $|b_{n}(t)|\leq\eta^{-n}\sup_{B(0,\eta)}|g_{t}|$ and that
$|g_{t}(z)|$ is uniformly bounded for $(z,t)$ small enough we get that
$t\mapsto f_{t}(z)$ is for each fixed $z$ a convergent series of holomorphic
functions of $t$, hence holomorphic.

Then we take $R_{t}(z)=\log\frac{f_{t}(z)}{\operatorname{res}_{t}+1}$. For
$\eta$ and $r^{\prime}$ small enough the map $f_{t}$ will be non-vanishing and
we can choose the branch of the log to be analytically varying with $(z,t)$.
It follows that $w=z\exp(R_{t}(z)/(\operatorname{res}_{t}+1))$ is holomorphic
in $t$.

For $t=0$, $z\mapsto w$ it is the restriction of a change of variable defined
on $B(0,\varepsilon)$. By a theorem of Hurwitz, it will still be injective on
$B(0,\eta)$ for $|t|<r^{\prime}$ provided $r^{\prime}$ is small enough.

A.2. Following the resting point: Proof of Lemma 7

Holomorphy is a local property and by a change of parameter it is enough to
prove the claim on a subset $B(0,r^{\prime})\subset B(0,r)$ of values of $t$.
Let $B=B(z_{1}(0),R)$ with $R$ small enough so that $\overline{B}\subset
U-\\{z_{0}(0)\\}$. Choose also $R$ small enough so that we can apply Lemma 5
to get a special coordinate $z\mapsto w$ on $B$ for the singularity $z_{1}(t)$
provided $t$ is small enough. Choose181818Project $\gamma_{0}(s)$ radially to
the boundary of a slightly bigger disk, until $s$ is big enough that the rest
of the path is inside and then replace by a radial semgent: one gets a
homotopic path. a representative $\gamma_{0}$ of $[\gamma_{0}]$ that reaches
$\partial B$ for the first time for some $s=s_{1}$ and then goes down to
$z_{1}$ along a radial segment of $B$. In particular the ball $B(z_{1}(0),R)$
is disjoint from $\gamma_{0}([0,s_{1}])$.

The resting place $c$ for small $t$ will be computed by patching a finite
number of local solutions $f$ of $f^{\prime\prime}/f^{\prime}=\zeta_{t}$ as
follows. Choose $\eta>0$. For $N\geq 2$ big enough the intervals
$[s_{1}k/N,s_{1}(k+1)/N]$, $0\leq k\leq N-1$, are mapped by $\gamma_{0}$ to
subsets of the balls $B_{k}=B(u_{k},\eta)$ where for convenience we denote

$u_{k}=\gamma_{0}\left(s_{1}\frac{k}{N}\right).$

For $\eta$ small enough these balls are contained in $U$. Note that
$[0,s_{1}]\subset[0,1)$. Choose some $\varepsilon>0$ so that
$\varepsilon<\min(\eta,\varepsilon<R/2,\varepsilon_{0})$, where
$\varepsilon_{0}$ is the $\varepsilon$ given by Lemma 22 applied to
$\gamma_{0}$. For $t$ small enough, $X_{t}$ belongs to the neighbourhood
$V(\gamma,\varepsilon)$ of $X_{0}$: $z_{0}(t)\in B(z_{0}(0),\varepsilon)$,
$z_{1}(t)\in B(z_{1}(0),\varepsilon)$, and by Lemma 22, $\gamma_{t}$ is
homotopic to $\gamma=\delta\cdot(\psi\circ\gamma_{0})$ for _any_ path $\delta$
from $\tilde{z}_{0}(t)$ to $z_{0}(0)$ within $B(z_{0}(0),\varepsilon)$, and
_any_ homeomorphism $\psi$ of $U$ that maps $z_{1}(0)$ to $z_{1}(t)$ and which
is the identity outside $B(z_{1}(0),\varepsilon)$. We choose
$\delta=\delta_{t}$ to be a straight segment and $\psi_{t}$ to send the rays
$[z_{1}(0),u]$ of the disk
$\overline{B}^{\prime}=\overline{B}(z_{1}(0),\varepsilon)$ to the segments
$[z_{1}(t),u]$ and let

$\gamma_{t}:=\delta_{t}\cdot(\psi_{t}\circ\gamma_{0})$

In partiuclar $\psi_{t}\circ\gamma(s)$ follows the straight segment
$[\gamma_{0}(s_{1}),z_{1}(t)]$ for $s\in[s_{1},1]$. For a fixed $t$, for all
$0\leq k\leq N-1$ let inductively $\phi_{k,t}$ denote the solution of
$\phi^{\prime\prime}_{k,t}/\phi^{\prime}_{k,t}=\zeta_{t}$ on $B_{k}$ such that

  * $-$

the germ of $\phi_{0,t}$ at $z_{0}(t)$ is $\phi_{t}$,

  * $-$

for $k\geq 1$, $\phi_{k,t}$ and $\phi_{k-1,t}$ coincide in a neighbourhood of
$u_{k}$.

The maps $\phi_{k,t}$, as solutions $f$ of
$f^{\prime\prime}/f^{\prime}=\zeta_{t}$ with initial conditions $f(u)=v$ and
$f^{\prime}(u)=w$, depend holomorphically on $t$, $u$, $v$ and $w$. It follows
by induction that the points $\phi_{k,t}(u_{k})$, $\phi_{k,t}(u_{k+1})$ and
the derivatives $\phi_{k,t}^{\prime}(u_{k})$, $\phi_{k,t}^{\prime}(u_{k+1})$,
for $0\leq k\leq N-1$, are holomorphic functions of the data.

For the last patch recall that $\psi_{t}\circ\gamma$ is a straight segment
from $\gamma_{0}(s_{1})$ to $z_{1}(t)$. We use a branch of $z\mapsto
w^{\operatorname{res}_{t}+1}$ on a straight sector $S$ in $z$ coordinate with
apex $z_{1}(t)$ and containing the segment $[u_{N},z_{1}(t))$, post-composed
with an appropriate ${\mathbb{C}}$-affine map $\sigma_{t}$ so as to match with
$\phi_{N-1,t}$ in a neighbourhood of $u_{N}$. The map $\sigma_{t}$ depends
holomorphically on the data and $c=\sigma_{t}(0)$.

A.3. Following saddle connections: proof of Propositions 8 and 9

$U$$z^{*}_{1}$$z^{*}_{0}$saddle connection Sketch in Riemann coordinates for
$\zeta^{*}$. $w_{1}^{*}$$w_{0}^{*}$$W$$L(t_{0})$$L(t_{1})$ Picture in
developped similarity coordinates, before perturbation. $W$ and the two
sectors are contained in the domain of $\psi^{*}$. If the perturbation is
small enough, there is still a segment between the sector apexes in the domain
of $\psi$. The sectors have scaled and rotated but the central bar has not
changed. Figure 8. Objects in the proof of Proposition 8. The sector
boundaries are not glued together (they are strict subsets of the sectors
defining a neighbourhood of the vertices).

We will say that $\zeta$ depends analytically on $\tau$ whenever $h$,
$\operatorname{res}_{k}$, $z_{k}$ do.

Let us call $\varepsilon$-close (to $\zeta^{*}$) the maps $\zeta$ as in the
statement of Proposition 8. For $\varepsilon$ small enough the two
singularities remain of residue of real part $>-1$, i.e. bounded.

Since geodesics go straight with constant speed in affine charts and since the
extent of times on which $f$ is defined is finite, by pasting inverses of
local charts, and of those of Lemma 1 at the extremities, one can prove that
there exists an open segment $(w_{0}^{*},w_{1}^{*})\subset{\mathbb{C}}$, a
neighbourhood $V$ of it, and a map $\psi^{*}:V\to U-\\{z^{*}_{0},z^{*}_{1}\\}$
such that

  * $-$

for all $t\in(0,1)$, $\psi^{*}((1-t)w_{0}^{*}+tw_{1}^{*})=f(t)$,

  * $-$

$\psi^{*}$ is analytic, locally invertible with local inverses being
similarity charts for $\zeta^{*}$,

  * $-$

$V$ contains a sector based on each end of $[w_{0}^{*},w_{1}^{*}]$ and having
this segment as a symmetry axis (see also Lemma 4).

Since $f$ is not necessarily injective, $\psi^{*}$ is not either.

Let

$L(t)=(1-t)w_{0}^{*}+tw_{1}^{*}.$

By a compactness argument, for all compactly contained open subset $W$ of $V$
containing the middle point $M=L(1/2)=(w_{0}^{*}+w_{1}^{*})/2$, there is
$\varepsilon>0$ such that a map $\psi$ persists on $W$ for all
$\varepsilon$-close $\zeta$ such that,

  * $-$

$\psi(M)=\psi^{*}(M)$, $\psi^{\prime}(M)=(\psi^{*})^{\prime}(M)$,

  * $-$

$\psi$ is analytic, locally invertible with local inverses being similarity
charts for $\zeta$.

Moreover, $\psi$ depends analytically on $\tau$ if $\zeta$ does. This comes
from the expression of local similarity charts in terms of $\zeta$ (or as a
variant: from the fact that $\psi$ is a solution of the complex ODE
$-\psi^{\prime\prime}/\psi^{\prime 2}=\zeta\circ\psi$.)

Near singularities, according to Lemma 5, the formula for the holomorphic
change of variable $z\mapsto w$ in Lemma 1 depends holomorphically on $\tau$
if $\zeta$ does. Recall that branches of $w^{1+\operatorname{res}}$ are
similarity charts. We can thus choose for $\varepsilon$ small enough some
times $t_{0}$ close to $0$ and $t_{1}$ close to $1$, independent of $\zeta$,
such that there are similarity chart inverses $\psi_{0}$, $\psi_{1}$ defined
on the sector

$\big{\\{}{u=re^{i\theta}}\,\big{|}\,{0<r<2,\ -\eta<\theta<\eta}\big{\\}}$

for some $\eta$ independent of $\zeta$, and such that, as $u\to 0$,
$\psi_{0}(u)\longrightarrow z_{0}$, $\psi_{1}(u)\longrightarrow z_{1}$,
$\psi_{0}(1)=\psi(L(t_{0}))$ and $\psi_{1}(1)=\psi(L(t_{1}))$.

Choose $W$ as above to contain the sub-segment $[L(t_{0}),L(t_{1})]$ of
$[w^{*}_{0},w^{*}_{1}]$. We can patch together $\psi$, $\psi_{0}$ and
$\psi_{1}$ by replacing $\psi_{i}$ with
$\tilde{\psi}_{i}:w\mapsto\psi_{i}(s_{i}(w))$ where $s_{i}$ is is the
${\mathbb{C}}$-affine map such that
$\tilde{\psi}_{i}(L(t_{i}))=\psi(L(t_{i}))$ (i.e. $s_{i}(L(t_{i}))=1$),
$\tilde{\psi}_{i}^{\prime}(L(t_{i}))=\psi^{\prime}(L(t_{i}))$, and possibly
reducing $\eta$ and $W$. We obtain an extended map $\tilde{\psi}$ defined on
the union of a fixed neighbourhood $W$ of $[L(t_{0}),L(t_{1})]$ and of two
sectors centred on points $w_{0}$ and $w_{1}$ that depend on $\zeta$, of
symmetry axis $[w_{i},L(t_{i})]$, of opening angle $\eta$ independent of
$\zeta$, and of radius $2|w_{i}-L(t_{i})|$. See Figure 8.

Everything in this construction depends holomorphically on $\tau$ if $\zeta$
does. For $\varepsilon$ small enough, the segment $[w_{0},w_{1}]$ (which
depends on $\zeta$) is contained in the domain of $\tilde{\psi}$ (which also
depends of $\zeta$) and we can take $f(t)=\tilde{\psi}((1-t)w_{0}+tw_{1})$ as
a map satisfying the conclusion of Proposition 8.

$w^{*}_{0}$$w^{*}_{1}$$L(t_{0})$$L(t_{1})$ Figure 9. Adaptation of Figure 8
for the proof of Proposition 9

The proof of Proposition 9 is very similar, with one of the sectors replaced
by an unbounded sector, using Lemmas 4 and 6 for this part. See Figure 9.

Appendix B Example of a hole type infinite vertex

Consider the strip $P:``{\operatorname{Re}\,}z\in[1,2]"$ and pair the left
unbounded edge with the right one by sending $z\mapsto 2z$, i.e. $1+iy\to
2+2iy$. (To respect the temporary restrictions of the section _Unbounded
polygons_ on page 2 one can cut $P$ in two pieces along ${\mathbb{R}}$ and
pair the two bounded edges together.) Then the quotient $\cal S$ is
homeomorphic to a sphere and there are two vertices, both unbounded. It is
then easy to uniformize the Riemann surface $\cal S^{\prime}$: it is
isomorphic to $H/\sim$ where $H$ is the half plane
“${\operatorname{Re}\,}z>0$” and $z\sim 2z$. Mapping the situation by
$z\mapsto\log z$ the set $H$ becomes the infinite band
“$|{\operatorname{Im}\,}z|<\pi/2$” and the equivalence relation becomes $z\sim
z+\log 2$. See Figure 10. Hence the Riemann surface $\cal S^{\prime}$ is
isomorphic to a cylinder of finite height, i.e. to a round annulus via the
transformation $z\mapsto e^{2\pi iz/\log 2}$, and both vertices are holes.

$z$$2z$…$\frac{1}{2}$$1$$2$$4$$\overset{\log}{\longrightarrow}$$\pi$$z$$z+\log
2$ Figure 10. The construction of Section 4.4. The gray fundamental domain is
glued via $s:z\mapsto 2z$. The quotient is isomorphic as a Riemann surface to
$H/s$ and via the complex logarithm to a strip modulo $z\mapsto z+\log 2$.

Appendix C Proof of holomorphic dependence (Proposition 11)

This appendix introduces a lot of notations, and we apologize in advance to
the reader. However, most, if not all, will only be used here.

Adding the image of the marked points

For each paired unbounded edges there was a marked point on the interior of
each. We extend the sequence $z_{k}$ by appending the image by $F:\cal
S\to\hat{\mathbb{C}}$ of the marked points, to get an (injective) element of
$\hat{\mathbb{C}}^{m}$ for some $m\geq p$ and we will prove that this extended
sequence depends holomorphically on the polygons.

Summary of the proof

Before entering in a fully detailed argument, let us give an overview of the
proof.

The construction of Proposition 11 starts from a collection of polygons and
gluings. We will define a space $\mathrm{S\text{-}Conf}$ of collections of
strictly convex polygons up to ${\mathbb{C}}$-affine transform, which
represent the possible geometric shapes that can take each of the polygons
when they vary while remaining strictly convex. (For unbounded polygons,
marked points are taken into account.) The space $\mathrm{S\text{-}Conf}$ is a
complex manifold, on which the construction can be carried out and yields in
particular the points $z_{k}\in\hat{\mathbb{C}}$ and their associated residues
$\operatorname{res}_{k}$. We already know by Lemma 10 that the
$\operatorname{res}_{k}$ depend holomorphically on the polygons and marked
points.

In fact we will see that one can get from the construction not only the
position of the points $z_{k}$ (up to an automorphism of $\hat{\mathbb{C}}$)
but an element of the Teichmüller space $\cal T$ of the Riemann sphere
$\hat{\mathbb{C}}$ with $m$ marked points. So we consider the map

$\operatorname{Glu}:\mathrm{S\text{-}Conf}\to\cal{TR}$

that sends a collection to the Teichmüller space element thus constructed,
together with the collection of residues. The name of the map reflects the
fact that it is defined by a process that starts by _gluing_ polygons. A
priori we do not know anything on $\operatorname{Glu}$, and even its
continuity should not be considered as a trivial statement.

To prove analytic dependence of the $z_{k}$ we will prove the stronger
statement that $\operatorname{Glu}$ is analytic. For this, we will introduce
sort of a reciprocal map

$\operatorname{Per}:\cal{TR}\to\mathrm{Conf}$

where $\mathrm{Conf}$ is a space containing $\mathrm{S\text{-}Conf}$. The name
$\operatorname{Per}$ is an abbreviation of _periods_. It is defined as
follows. An element of $\cal{TR}$ consists in an element of $\cal T$, in
particular points $z_{k}$, and the specification of complex numbers
$\operatorname{res}_{k}$ associated to each $z_{k}$ with some compatibility
conditions. One can associate the unique Christoffel symbol having these
residues at the $z_{k}$, and we thus get a similarity surface structure on
$\hat{\mathbb{C}}-\\{z_{k}\\}_{k=1}^{n}$. To the element of $\cal T$
corresponds a decomposition of the sphere into topological polygons, unique up
to isotopy of the sphere fixing the marked points. Local straightening maps of
the Christoffel symbol can be extended to these topological polygons into
holomorphic maps, which are not necessarily anymore injective. However, one
can look at the image of those topological polygons by these extended
straightening maps, and the position of the vertices in these images, up to a
${\mathbb{C}}$-affine map, is independent of the isotopy. This collection of
positions, up to ${\mathbb{C}}$-affine maps, is what $\operatorname{Per}$
associates to the element of $\cal{TR}$ we started from.

The name _periods_ of the map has been chosen because the relative position
can be computed by integrating an O.D.E. along paths between the $z_{k}$
(those that are bounded vertices or marked point) and the result only depends
on the homotopy class of the path; this is akin to computing translation
vectors between branch points of a translation surface by integration of a
one-form, and those vectors are sometimes called periods by extension of the
notion of period of a one-form along a closed loop.

One immediately gets that

$\operatorname{Per}\circ\operatorname{Glu}=\operatorname{Id}_{\mathrm{S\text{-}Conf}}.$

The space $\mathrm{Conf}$ is not an analytic manifold but we identify a subset
$\mathrm{Conf}^{*}$ that carries an analytic manifold structure and such that

$\mathrm{S\text{-}Conf}\subset\mathrm{Conf}^{*}\subset\mathrm{Conf},$

$\mathrm{S\text{-}Conf}$ is an open subset of $\mathrm{Conf}^{*}$ and their
analytic structures match. We then check in Lemma 16 that $\operatorname{Per}$
is continuous on $\cal{TR}$, and analytic on
$\operatorname{Per}^{-1}(\mathrm{Conf}^{*})$.

Let

$\mathrm{Eff}:=\operatorname{Glu}(\mathrm{S\text{-}Conf})\subset\cal{TR}.$

The name $\mathrm{Eff}$ is an abbreviation for _effective_. From
$\operatorname{Per}\circ\operatorname{Glu}=\operatorname{Id}_{\mathrm{S\text{-}Conf}}$
it immediately follows that

$\operatorname{Glu}\circ\operatorname{Per}|_{\mathrm{Eff}}=\operatorname{Id}_{\mathrm{Eff}}.$

Intuition tells us that $\mathrm{Eff}$ must be an open subset of $\cal TR$ but
this is a non-trivial fact. Recall that at this point of the proof, we do not
even know that $\operatorname{Glu}$ is continuous, so $\mathrm{Eff}$ could
just be be any kind of uncountable subset of $\cal TR$. On the other hand, it
is immediate that the subset $\operatorname{Per}^{-1}(\mathrm{S\text{-}Conf})$
of $\cal{TR}$ is open, since $\operatorname{Per}$ is holomorphic hence
continuous.

A key point (see below), which will require some effort, will be to prove that
each point in $\mathrm{Eff}$ has a neighbourhood $W$ in
$\operatorname{Per}^{-1}(\mathrm{S\text{-}Conf})$, on which the equality

$\operatorname{Glu}\circ\operatorname{Per}|_{W}=\operatorname{Id}_{W}$

holds. This implies that $W\subset\mathrm{Eff}$ and proves that $\mathrm{Eff}$
is an open subset of $\cal{TR}$.

To conclude, we use a theorem that if one has an analytic map $\phi:U\to V$
between open subsets of complex manifolds of dimensions $a$ and $b$, and if
$\phi$ is a bijection, then $a=b$ and $\phi^{-1}$ is analytic too. We apply
this to $\phi=\operatorname{Per}|_{\mathrm{Eff}}:U=\mathrm{Eff}\to
V=\mathrm{S\text{-}Conf}$.

The aforementioned key point follows from Lemma 17 and the paragraph after it.
An effective element $x$ of $\cal{TR}$ corresponds to a collection of strictly
convex polygons $P_{j}$. Nearby elements $x^{\prime}$ of $\cal{TR}$ will map
by $\operatorname{Per}$, which is continuous, to configurations of points for
which there exists strictly convex polygons $P^{\prime}_{j}$ having these
points as vertices and marked points. The difficulty is to prove that gluing
these new polygons indeed gives back $x^{\prime}$. For this we follow the
saddle connections as a function of $x^{\prime}$. This defines a cellular
decomposition of the sphere and we check that the cells actually map to the
$P^{\prime}_{j}$ by the straightening maps of $x^{\prime}$, provided it is
close to $x$. Since the cellular decomposition for $x^{\prime}$ is obtained by
deformation of the one for $x$, we can check that gluing the $P^{\prime}_{j}$
also gives the right isotopy class for the Teichmüller part of $x^{\prime}$.

Polygon space

Consider strictly convex bounded polygons $P\subset{\mathbb{C}}$: they are
assumed with non-empty interior and have at least three vertices. Let us
orient the boundary of $P$ in the counter clockwise way, and index its
vertices accordingly as follows: $w_{1},\ldots,w_{n}$. The set of strictly
convex polygons with indexed vertices thus forms an open subset of
${\mathbb{C}}^{n}$.

For the unbounded polygons, recall that we assumed: that they are strictly
convex; that there are exactly two unbounded edges; that they make a positive
angle (this is a supplementary assumption made in Proposition 11); that we
consider $\infty$ as a vertex; that we include a marked point in the interior
of each of the two unbounded edges. We still orient the boundary in the anti-
clockwise way and index the vertices and marked points accordingly _but
omitting $\infty$_: $w_{1},\ldots,w_{n}$. We still get an open subset of
${\mathbb{C}}^{n}$, where now $n$ is the number of edges plus one, so $n\geq
3$.

Quotienting the polygon space

Changing one polygon by mapping it under an affine map does not change the
quotient surfaces $\cal S$, $\cal S^{\prime}$ (there are trivial canonical
isomorphisms), nor the sequence $(z_{k})$, hence we can work in the space of
(possibly marked) polygons $P_{j}$ up to the action of affine maps: this is
still a complex manifold (it amounts to fixing the affix of two particular
vertices/marked points of $P_{j}$) and the quotient map is analytic and open.

In fact we will use later larger spaces. Let $\operatorname{Aff}{\mathbb{C}}$
denote the group of ${\mathbb{C}}$-affine maps $w\mapsto aw+b$ of
${\mathbb{C}}$ and let it act on ${\mathbb{C}}^{n}$ component-wise: for
$(w_{k})\in{\mathbb{C}}^{n}$, $s\cdot(w_{k}):=(s(w_{k}))$. Denote

$A{\mathbb{C}}^{n}={\mathbb{C}}^{n}/\operatorname{Aff}{\mathbb{C}}.$

The quotient map ${\mathbb{C}}^{n}\to A{\mathbb{C}}^{n}$ is open for the
quotient topology.191919I.e. the saturate of an open subset of
${\mathbb{C}}^{n}$ is open, which is easily checked here since affine maps are
open. Let $A^{*}{\mathbb{C}}^{n}\subset A{\mathbb{C}}^{n}$ be the quotient of
${\mathbb{C}}^{n}-\Delta$ where
$\Delta=\big{\\{}{(w,\ldots,w)}\,\big{|}\,{w\in{\mathbb{C}}}\big{\\}}$ i.e.
$\Delta$ represents the case where all points have merged into a single point.
The space $A^{*}{\mathbb{C}}^{n}$ holds a complex manifold structure such that
the quotient map ${\mathbb{C}}^{n}-\Delta\to A^{*}{\mathbb{C}}^{n}$ is
analytic. Locally, charts of $A^{*}{\mathbb{C}}^{n}$ are given by fixing the
affix of two distinct points. The space $A{\mathbb{C}}^{n}$ has only one point
more than $A^{*}{\mathbb{C}}^{n}$ and we do not seek to extend the complex
structure to this point. For the quotient topology $A{\mathbb{C}}^{n}$ is not
Hausdorff separated: indeed the special point has only one neighbourhood,
which is the whole space $A{\mathbb{C}}^{n}$ (this is sometimes called a
_focal point_).

Recall that we indexed the collection of polygons $P_{j}$ by a finite set $J$.
Let $n_{j}$ be the number of vertices on $P_{j}$ (including marked points,
excluding infinity). Let

$\displaystyle\mathrm{Conf}=\prod_{j\in J}A{\mathbb{C}}^{n_{j}}$
$\displaystyle\mathrm{Conf}^{*}=\prod_{j\in J}A^{*}{\mathbb{C}}^{n_{j}}$

For each $j$, the number $n_{j}$ is fixed and we also fix the set of $j$ for
which the $w_{k}$ include marked points, whose indexes we fix and must be two
integers that are consecutive modulo $n_{j}$. Let
$\mathrm{S\text{-}Conf}\subset\mathrm{Conf}$ denote the elements such that for
each $j$ without marked point, the $(w_{k})$ are given by the anti-clockwise
indexed vertices of a strictly convex bounded $P_{j}$, and in the case with
marked point, the $(w_{k})$ are given as explained in the paragraph above
called “Polygon space”. Then the subset $\mathrm{S\text{-}Conf}$ of
$\mathrm{Conf}$ is open and

$\mathrm{S\text{-}Conf}\subset\mathrm{Conf}^{*}\subset\mathrm{Conf}.$

As a product of analytic manifolds, $\mathrm{Conf}^{*}$ is an analytic
manifold and this makes $\mathrm{S\text{-}Conf}$ an analytic manifold too.

Moduli space and Teichmüller space of the sphere with marked points

We fix a particular polygon configuration in $\mathrm{S\text{-}Conf}$, which
we will call $[\cal P^{0}]$. Denote $\cal S^{0}$, $\cal
V^{0}=\big{\\{}{v_{k}^{0}}\,\big{|}\,{1\leq k\leq m}\big{\\}}$, and $(\cal
S^{\prime})^{0}=\cal S^{0}-\cal V^{0}$ the respective Riemann surface,
vertices, and similarity surface associated to $\cal P^{0}$.

A quick reminder on the moduli and Teichmüller spaces of the sphere with
marked points can be found in Section 4.4. However, for the reader’s
convenience, we copy here some of the definitions given there.

The moduli space is the subset $\cal M\subset\hat{\mathbb{C}}^{n}$ of
$n$-uplets $(z_{k})\in\hat{\mathbb{C}}^{n}$ for which the $z_{k}$ are distinct
and $z_{1}=0$, $z_{2}=1$, $z_{3}=\infty$. It is a complex manifold of
dimension $n-3$. To a configuration $\cal P\in\mathrm{S\text{-}Conf}$ we
associate the image in $\hat{\mathbb{C}}$ of its vertices and marked points:
$Z=(z_{k})=(F(v_{k}))\in\cal M$ for the associated conformal isomorphism
$F:\cal S\to\hat{\mathbb{C}}$ sending $v_{1}$ to $0$, $v_{2}$ to $1$ and
$v_{3}$ to $\infty$ (as defined before Proposition 11), and we let $\cal
Z=\big{\\{}{z_{k}}\,\big{|}\,{1\leq k\leq n}\big{\\}}$. To $\cal P$ we also
associate the Christoffel symbol $\zeta_{\cal P}$ on $\hat{\mathbb{C}}-\cal Z$
that our construction yields. Naturally, we denote $F^{0}$, $Z^{0}$ and $\cal
Z^{0}$ the objects associated to $\cal P^{0}$.

Let $\cal T$ be the Teichmüller space associated to $\cal S^{0},\cal V^{0}$
defined (without the use of quasiconformal maps) as follows:

$\cal T=\cal F/\cal H^{0}$

where $\cal F$ is the set of orientation preserving homeomorphisms $f:\cal
S^{0}\to\hat{\mathbb{C}}$ sending respectively $v^{0}_{1}$, $v^{0}_{2}$ and
$v^{0}_{3}$ to $0$, $1$ and $\infty$ and $\cal H^{0}$ is the set of
orientation preserving self homeomorphisms of $\cal S^{0}$ that are isotopic
to the identity rel. $\cal V^{0}$, i.e. by an isotopy fixing each vertex
$v_{k}^{0}$.

If $f\in\cal F$, its equivalence class modulo right composition with $\cal
H^{0}$ will be denoted $[f]\in\cal T$. Hence, $[f_{1}]=[f_{2}]$ iff
$\exists\phi\in\cal H^{0}$ such that $f_{2}=f_{1}\circ\phi$. But note that
this is equivalent to: $f_{2}$ is isotopic to $f_{1}$ rel. $\cal
Z=\\{z_{k}=f_{1}(v_{k}^{0})\\}$.

Let us endow $\hat{\mathbb{C}}$ with a spherical metric202020A metric for
which the circles are geometric circles via a stereographic projection. $d$
and the set $\cal F$ with the metric

$d(f_{1},f_{2})=\sup\big{\\{}{d(f_{1}(x),f_{2}(x))}\,\big{|}\,{x\in\cal
S^{0}}\big{\\}}.$

The space $\cal T$ is endowed with the quotient topology. It is separated
(distinct points have some disjoint neighbourhoods). The projection $\Pi:\cal
F\to\cal T$ is open and has continuous local sections (it is a fibre bundle,
see Section 4.4). Since $\Pi$ is open, a subset of $\cal T$ is a neighbourhood
of $[f]\in\cal T$ iff for some $\varepsilon>0$ it contains the set of
$[f^{\prime}]$ for which $d(f,f^{\prime})<\varepsilon$. We will use the
following topological lemma. Endow $\cal S^{0}$ with a distance $d$ inducing
its topology and $\cal H^{0}$ with the the distance

$d(\phi_{1},\phi_{2})=\sup\big{\\{}{d(\phi_{1}(x),\phi_{2}(x))}\,\big{|}\,{x\in\cal
S^{0}}\big{\\}}.$

Then (see Section 4.4, Lemma 29):

######  Lemma 13.

There exists $\varepsilon>0$ such that for every orientation preserving self-
homeomorphism $\phi$ of $\cal S^{0}$, if $\phi$ fixes every point in $\cal
V^{0}$ and satisfies $d(\phi,\operatorname{id}_{\cal S^{0}})<\varepsilon$ then
$\phi\in\cal H^{0}$ (i.e. $\phi$ is isotopic to the identity rel. $\cal
V^{0}$).

Stating the objective in terms of analyticity of a function
$\operatorname{Glu}$

$P_{1}^{0}$$P_{2}^{0}$$P_{3}^{0}$$\cal P^{0}$$P_{1}$$P_{2}$$P_{3}$$\cal
P$$\cal S^{0}$$\cal
S$$z_{k}^{0}$$z_{k}$$\pi^{0}$$\pi$$F^{0}$$F$$\phi$$\psi$$f$ Figure 11. This
commuting diagram sums up the objects involved in the construction of
$\operatorname{Glu}$. For a fixed $\cal P$, the descending maps are not unique
but $\pi$, $F$ and $[f]$ are.

The different objects that will be introduced and their relation are summed up
in Figure 11.

For a polygon configuration in $\mathrm{S\text{-}Conf}$, let us explain why
our gluing construction not only gives us an element of $\cal M$ but in fact
an element of $\cal T$. For this, given a polygon configuration $[\cal
P]\in\mathrm{S\text{-}Conf}$ we build below a homeomorphism $\psi:\cal
S^{0}\to\cal S$. It is not unique but its class modulo $\cal H^{0}$ will be
unique.

Consider a collection $\phi=(\phi_{j})$ of homeomorphisms $\phi_{j}:\partial
P^{0}_{j}\to\partial P_{j}$ that send vertices and marked points of
$P^{0}_{j}$ to their corresponding points in $P_{j}$ (in particular, they are
orientation preserving) and such that the $\phi_{j}$ are compatible with the
identification of paired edges: if $(e\,{:}\,j)$ and
$(e^{\prime}\,{:}\,j^{\prime})$ are paired in $\cal P^{0}$, and $s$ is the
${\mathbb{C}}$-affine map matching them, calling $(\tilde{e}\,{:}\,j)$,
$(\tilde{e}^{\prime}\,{:}\,j^{\prime})$ and $\tilde{s}$ the corresponding
objects in $\cal P$, we ask that $\phi_{j^{\prime}}\circ
s=\tilde{s}\circ\phi_{j}$. Such a collection exists, but is not unique. Then
$\phi_{j}$ can be extended to an orientation preserving homeomorphism
$\phi_{j}:P^{0}_{j}\to P_{j}$. Let us join all these maps into a map
$\phi:\bigsqcup_{j}P^{0}_{j}\to\bigsqcup_{j}P_{j}$. This join descends to an
orientation preserving homeomorphism $\psi:\cal S^{0}\to\cal S$ that depends
only on $\phi$ such that $\pi\circ\phi=\psi\circ\pi^{0}$, where

$\pi:\bigsqcup_{j}P_{j}\to\cal S\text{ and
}\pi^{0}:\bigsqcup_{j}P^{0}_{j}\to\cal S^{0}$

are the projection to the respective quotients. This is summed up in the
following commutative diagram:

${\bigsqcup_{j}P^{0}_{j}}$${\bigsqcup_{j}P_{j}}$${\cal S^{0}}$${\cal
S}$$\scriptstyle{\phi}$$\scriptstyle{\pi^{0}}$$\scriptstyle{\pi}$$\scriptstyle{\psi}$

Different choices for $\phi$ yield different maps $\psi$ but they are all
equivalent modulo $\cal H^{0}$.212121This is directly seen to be equivalent
to: the maps $\psi^{-1}$ are all isotopic rel. the vertices and marked points
of $\cal S^{0}$. Actually by taking inverse this is also equivalent to stating
that the maps $\psi$ are isotopic rel the vertices and marked points of $\cal
S$. This follows from the following topological facts, via a homeomorphism
from $P^{0}_{j}$ to $\overline{{\mathbb{D}}}$: the set of orientation
preserving self-homeomorphisms of a segment is connected (it is convex!), an
isotopy of the circle can be radially extended to an isotopy of the closed
unit disk $\overline{{\mathbb{D}}}$ (immediate), the set of self-
homeomorphisms of $\overline{{\mathbb{D}}}$ that are the identity on
$\partial{\mathbb{D}}$ is connected (easy thanks to Alexander’s trick
[Ale24]).

Recall that $F$ denotes a specific Riemann surface isomorphism from $\cal S$
to $\hat{\mathbb{C}}$. Then the map

$f:=F\circ\psi$

is an element of $\cal F$. (By what we proved in the previous paragraph,) its
class modulo pre-composition by elements of $\cal H^{0}$ is independent of the
choice of $\phi$ and is the element of $\cal T$ we associate to $\cal P$. This
independence is used in the proof of Lemma 17.

We partition the index set $\\{1,\ldots m\\}$ into three subsets as follows:
let $F\subset\\{1,\ldots m\\}$ denote the set of those indices $k$ for which
the vertex $v^{0}_{k}$ of the base surface $\cal S^{0}$ is bounded,
$I\subset\\{1,\ldots m\\}$ those for which it is unbounded and
$M\subset\\{1,\ldots m\\}$ be the marked points. Let

(7) $\cal
R=\left\\{{(r_{k})\in{\mathbb{C}}^{m}}\,\middle|\,{\begin{array}[]{c}\sum
r_{k}=-2\\\ \forall k\in F,\ {\operatorname{Re}\,}r_{k}>-1\\\ \forall k\in I,\
{\operatorname{Re}\,}r_{k}<-1\\\ \forall k\in M,\
r_{k}=0\end{array}}\right\\}$

and

$\cal{TR}=\cal T\times\cal R.$

To a polygon configuration $[\cal P]\in\mathrm{S\text{-}Conf}$ let us
associate the element $[f]$ of $\cal T$ that the construction above yields.
Let us also associate the set of residues $\operatorname{res}_{k}$ of the
Christoffel symbol $\zeta_{P}$ at the $m$ vertices/marked point and
temporarily denote

$\operatorname{res}=(\operatorname{res}_{k})\in{\mathbb{C}}^{m}.$

We saw earlier that $\sum\operatorname{res}_{k}=-2$, and the types of the
singularities (bounded or unbounded) of $\zeta$ are the same as for $\cal
S^{0}$ according to Section 3.2 and for marked points there is no singularity,
so $\operatorname{res}\in\cal R$. We denote

$\operatorname{Glu}:\left.\begin{array}[]{rcl}\mathrm{S\text{-}Conf}&\to&\cal{TR}\\\
{}[\cal P]&\mapsto&([f],\operatorname{res})\end{array}\right.$

where $f$ and $\operatorname{res}$ are defined above. We call
$\operatorname{Glu}$ the _gluing_ map. We want to prove that
$\operatorname{Glu}$ is analytic. But we do not even know yet that it is
continuous.

An analytic inverse candidate $\operatorname{Per}$

Consider any element $([f],\operatorname{res})$ of $\cal{TR}$, $f:\cal
S^{0}\to\hat{\mathbb{C}}$ and $(\operatorname{res}_{k})\in\cal R$. Let
$z_{k}=f(v_{k}^{0})$ be the affixes that any map in $[f]$ gives to the
vertices and let $\cal Z=\big{\\{}{z_{k}}\,\big{|}\,{1\leq k\leq n}\big{\\}}$
denote their set. Consider the unique Christoffel symbol $\zeta$ that is
holomorphic on $\hat{\mathbb{C}}-\cal Z$ and whose singularities are at most
simple poles and have residues $\operatorname{res}_{k}$ at $z_{k}$. By
definition of $\cal R$ if the type of $v_{k}^{0}$ is finite (F), infinite (I),
marked (M) then the corresponding singularity $z_{k}$ of $\zeta$ is
respectively bounded (${\operatorname{Re}\,}\operatorname{res}_{k}>-1$),
unbounded (${\operatorname{Re}\,}\operatorname{res}_{k}<-1$), erasable
(${\operatorname{Re}\,}\operatorname{res}_{k}=0$). Consider a polygon
$P_{j}^{0}$ of $\cal P^{0}$, choose any point $w_{*}$ in the interior of
$P^{0}_{j}$, and let $z_{*}=f\circ\pi^{0}(w_{*})$ (recall that
$\pi^{0}:\bigsqcup_{j}P^{0}_{j}\to\cal S^{0}$). Choose any germ $\phi_{0}$ of
similarity chart for $\zeta$ at $z_{*}$. For all bounded vertices and all
marked points $w$ of $P^{0}_{j}$, the germ can be developed along any path
within $f\circ\pi^{0}(P^{0}_{j})$ going to $z=f\circ\pi^{0}(w)$ without
hitting the vertices (except at the end) and the associated resting place (see
Section 3.6) is independent of the choice of the path since $P^{0}_{j}$ is
simply connected. Let

$R\in{\mathbb{C}}^{n_{j}}$

be the indexed collection of all these resting places.

######  Lemma 14.

The projection of $R$ in $A{\mathbb{C}}^{n_{j}}$ is independent of the
representative of $f$ in $[f]$, of the choice of $w_{*}$ and of the choice of
$\phi_{0}$.

Before proving it, let us gather all those projections for the different $j\in
J$: we have hence defined a map

$\operatorname{Per}:\cal{TR}\to\mathrm{Conf}$

that we call the _periods_ map.

To prove Lemma 14, and for further uses, we extend the definition of $R$.
Assume that $z_{*}\in\hat{\mathbb{C}}-\cal Z$ is any point, that $\phi_{0}$ is
a germ of straightening chart at $z_{*}$, that
$\gamma_{i}:[0,1]\to\hat{\mathbb{C}}$, $i\in\\{1,\ldots,p\\}$,
$p\in{\mathbb{N}}$ are paths such that $\gamma_{i}(0)=z_{*}$,
$\gamma_{i}([0,1))\subset\hat{\mathbb{C}}-\cal Z$ and $\gamma_{i}(1)\in\cal Z$
is a bounded singularity of $\zeta$ (i.e. corresponds to a $v_{k}^{0}$ that is
either a finite vertex or a marked point). Develop $\phi_{0}$ along each
$\gamma_{i}$ and call $c_{i}\in{\mathbb{C}}$ its resting point. Let
$R=(c_{i})_{i=1}^{p}$, which depends on all the choices above.

######  Lemma 15.

Consider two choices of $z_{*}$, $\phi_{0}$, $\gamma_{i}$ and
$z_{*}^{\prime}$, $\phi^{\prime}_{0}$ and $\gamma^{\prime}_{i}$ as above and
assume that there exists a path $\delta$ from $z_{*}^{\prime}$ to $z_{*}$ such
that each path $\gamma^{\prime}_{i}$ is homotopic to the concatenation
$\delta\cdot\gamma_{i}$ by a homotopy leaving fixed the starting and
endpoints, and avoiding $\cal Z$ except at the endpoint. Then the projections
of the corresponding $R$ and $R^{\prime}$ in $A{\mathbb{C}}^{p}$ are
identical.

###### Proof.

Let us develop $\phi^{\prime}_{0}$ along $\delta$. We obtain at the end a germ
$\tilde{\phi}_{0}$ at $z_{*}$. This germ takes the form
$\tilde{\phi}_{0}=s\circ\phi_{0}$ for some ${\mathbb{C}}$-affine map $s$. The
development of $s\circ\phi_{0}$ along $\gamma_{i}$ is the post-composition by
$s$ of the development of $\phi_{0}$ along the same path $\gamma_{i}$ and its
resting place is $s(c_{i})$. By the homotopy assumption and properties of
resting places (see Section 3.6) the resting places $c_{i}^{\prime}$ composing
$R^{\prime}$ are the same as the resting places obtained from
$z^{\prime}_{*}$, $\phi^{\prime}_{0}$, $\delta\cdot\gamma_{i}$. The
development of $\phi^{\prime}_{0}$ along $\delta\cdot\gamma_{i}$ is the
concatenation of the development of $\phi^{\prime}_{0}$ along $\delta$ and the
development of $\tilde{\phi}_{0}$ along $\gamma_{i}$. Hence the
$c^{\prime}_{i}$ are the same as the resting places obtained from $z_{*}$,
$s\circ\phi_{0}$, $\gamma_{i}$, i.e. $c^{\prime}_{i}=s(c_{i})$. ∎

###### Proof of Lemma 14.

Assume that $[f]=[f^{\prime}]$, that $w_{*}$ and $w_{*}^{\prime}$ are in the
interior of $\cal P^{0}_{j}$ and that $\phi_{0}$ and $\phi_{0}^{\prime}$ are
germs of straightening coordinates of $\zeta$ at these points. Let
$z_{*}=f(w_{*})$ and $z^{\prime}_{*}=f^{\prime}(w_{*}^{\prime})$. Consider
paths $\eta_{i}$, $i\in\\{1,\ldots,n_{j}\\}$, in $\cal P^{0}_{j}$ from $w_{*}$
to the finite vertices and marked points and a path $\beta$ from
$w^{\prime}_{*}$ to $w_{*}$ within $\cal P^{0}_{j}$. Let
$\gamma_{i}=f\circ\eta_{i}$. Let
$\gamma^{\prime}_{i}=f^{\prime}\circ(\beta\cdot\eta_{i})=(f^{\prime}\circ\beta)\cdot(f^{\prime}\circ\eta_{i})$.

Since $[f]=[f^{\prime}]$ there is an isotopy $s\in[0,1]\mapsto f_{s}$ from
$f^{\prime}$ to $f$ fixing the images of all vertices and marked
points.222222Write $f^{\prime}=f\circ h$ for some $h\in\cal H_{0}$, let
$h_{s}$ be a path from $h$ to $\operatorname{Id}$ in $\cal H^{0}$ and let
$f_{s}=f\circ h_{s}$. In particular $f_{0}=f^{\prime}$ and $f_{1}=f$. Let
$\alpha:\,t\in[0,1]\mapsto\alpha(t)=f_{t}(w_{*})$ which goes from
$\alpha(0)=f^{\prime}(w_{*})$ to $\alpha(1)=f(w_{*})=z_{*}$. Let $\delta$ be
the concatenation $(f^{\prime}\circ\beta)\cdot\alpha$. Then the conditions of
Lemma 15 are satisfied, with the following homotopy from
$\delta\cdot\gamma_{i}$ to $\gamma^{\prime}_{i}$: first
$f^{\prime}\circ\eta_{i}$ is homotopic to $\alpha\cdot\gamma_{i}$ via
$(s,t)\in[0,1]^{2}\mapsto\alpha(2t)$ if $t\leq s/2$ and
$f_{s}\left(\eta_{i}\left(\frac{t-\frac{s}{2}}{1-\frac{s}{2}}\right)\right)$
if $t\geq s/2$. Then
$\delta\cdot\gamma_{i}=((f^{\prime}\circ\beta)\cdot\alpha)\cdot\gamma_{i}$ is
homotopic to $(f^{\prime}\circ\beta)\cdot(\alpha\cdot\gamma_{i})$ hence to
$(f^{\prime}\circ\beta)\cdot(f^{\prime}\circ\eta_{i})=\gamma^{\prime}_{i}$.

The conclusion of Lemma 15 proves Lemma 14. ∎

Recall that we have put a complex manifold structure on $\mathrm{Conf}^{*}$
but not on $\mathrm{Conf}$.

######  Lemma 16.

The subset $\operatorname{Per}^{-1}(\mathrm{Conf}^{*})\subset\cal{TR}$ is open
and on it, the map $\operatorname{Per}$ is analytic.

###### Proof.

We will work locally so we consider an element
$(\tau^{\prime},\operatorname{res}^{\prime})\in\cal{TR}=\cal T\times\cal R$
such that
$\operatorname{Per}(\tau^{\prime},\operatorname{res}^{\prime})\in\mathrm{Conf}^{*}$,
and we will allow ourselves to restrict a finite number of times to smaller
and smaller neighbourhoods of this element. Let us focus on one polygon
$P^{0}_{j}$ and the collection of resting places
$R\in\hat{\mathbb{C}}^{n_{j}}$ used in the definition of $\operatorname{Per}$.

Fix any $f^{\prime}\in\cal F$ such that $\tau^{\prime}=[f^{\prime}]$. Fix a
point $w_{*}^{\prime}\in P_{j}^{0}$ and let

$z_{*}^{\prime}=f^{\prime}\circ\pi^{0}(w_{*}^{\prime})\in\hat{\mathbb{C}}.$

We recall that the collection $R\in{\mathbb{C}}^{n_{j}}$ is associated to
$([f],\operatorname{res})\in\cal{TR}$ and to a choice of point $w_{*}$ in the
interior of $P^{0}_{j}$ and of an initial germ $\phi_{0}$ of similarity chart
at $z_{*}=f\circ\pi^{0}(w_{*})$.

We want to keep $z_{*}$ fixed and equal to $z_{*}^{\prime}$, and hence we want
to define $w_{*}=(f\circ\pi_{0})^{-1}(z_{*}^{\prime})$. This poses several
problems. For one thing, $f$ is only defined up to isotopy rel. the vertices
and marked points, and thus $w_{*}$ depends on the choice of the
representative $f$ of $[f]$, and for some representatives, the point $w_{*}$
is not even in $P^{0}_{j}$. We could prove that the construction can be
carried out for such $w_{*}$ and work from that, but by simplicity we prefer
here to use that the map $\cal F\to\cal T:\ f\mapsto[f]$ has local sections
(Proposition 33 in Section 4.4). So we restrict to a neighbourhood $V$ of
$\tau^{\prime}$ in $\cal F$ on which there is a continuous map $V\to\cal
F:\tau\mapsto f_{\tau}$ with $[f_{\tau}]=\tau$. We then define

$w_{*}(\tau)=(f_{\tau}\circ\pi_{0})^{-1}(z_{*}^{\prime}).$

By making $V$ smaller, we ensure that $w_{*}(\tau)$ remains in the interior of
$P^{0}_{j}$. Note that $\tau\mapsto w_{*}(\tau)$ is continuous.

We fixed $z_{*}$. We also want to keep fixed the value and derivative at
$z_{*}$ of the initial germ used in the definition of $R$. Hence given any
point $(\tau,\operatorname{res})\in\cal{TR}$ such that $\tau\in V$, we choose,
as initial germ of straightening coordinate $\phi$ at $z_{*}$ for the
Christoffel symbol associated to $(\tau,\operatorname{res})$, the unique one
satisfying $\phi(z_{*})=\phi_{0}(z_{*})$ and
$\phi^{\prime}(z_{*})=\phi_{0}^{\prime}(z_{*})$. Note that
$z_{*}=z_{*}^{\prime}\neq\infty$, because $\infty$ is one of the vertices or
marked points by the convention that $z_{3}$ is always $\infty$.

Let us prove that $(\tau,\operatorname{res})\in V\times\cal R\mapsto
R(\tau,\operatorname{res})$ is analytic (recall that $\cal{TR}=\cal
T\times\cal R$ and that $\cal R$ is defined by eq. 7). We will take advantage
of a theorem of Hartogs that asserts that a function $F$ from a finite
dimensional complex manifold $X$ to another is analytic if and only if
$F\circ\xi$ is analytic for all one complex variable analytic map $\xi$ taking
values in $X$. The space $\cal{TR}$ is finite-dimensional. By Hartogs’
theorem, it is enough to prove that $t\mapsto
R(\tau_{t},\operatorname{res}(t))$ is analytic for all analytic map
$t\mapsto\tau_{t}\in V$ and all analytic map
$t\mapsto\operatorname{res}(t)\in\cal R$, where $t$ is a complex number.

Let us show how this follows from Lemma 7. Since
$R(\tau_{t},\operatorname{res}(t))$ is an element of ${\mathbb{C}}^{n_{j}}$,
we focus on one of its components. This component corresponds to some vertex
or marked point $v\in P^{0}_{j}$, more precisely it is the resting place of
any path $\gamma_{t}$ of the form
$\gamma_{t}=f_{\tau_{t}}\circ\pi^{0}\circ\delta_{t}$ where $\delta_{t}$ is a
path in $P^{0}_{j}$ from $w_{*}(\tau_{t})$ to $v$ that avoids the vertices and
marked points of $P^{0}_{j}$ except at the end. We choose $\delta_{t}$ to
depend continuously on $t$ (for instance a segment, linearly parametrized,
works since $P^{0}_{j}$ is convex and $w_{*}(\tau)$ is in its interior). Then
$\gamma_{t}$ depends continously on $t$.

Let $(z^{\prime}_{k})=M(\tau^{\prime})$, i.e.
$z_{k}^{\prime}=f^{\prime}(v^{0}_{k})$. Let $z^{\prime}_{k_{0}}$ be the one
that corresponds to the specific vertex we called $v$. The set $U$ on which we
will apply Lemma 7 is the complement in $\hat{\mathbb{C}}$ of the union of a
small closed disk around each $z^{\prime}_{k}$, except $z^{\prime}_{k_{0}}$.

The holomorphy hypotheses of Lemma 7 are immediately seen to be satisfied.
Continuity of $t\mapsto
X_{t}=(\gamma_{t}(0),\gamma_{t}(1),[\gamma_{t}])\in\Gamma$ follows from the
continuity of $t\mapsto\gamma_{t}$ and Lemma 26.

We have thus proved that the $R$, normalized as described earlier in this
proof, depends analytically on $(\tau,\operatorname{res})$ near
$(\tau^{\prime},\operatorname{res}^{\prime})$. Since we assumed that
$R(\tau^{\prime},\operatorname{res}^{\prime})$, which is an element of
${\mathbb{C}}^{n_{j}}$, does not have all its components equal, this is still
the case for nearby $(\tau,\operatorname{res})$. Hence
$\operatorname{Per}^{-1}(\mathrm{Conf}^{*})\subset\cal{TR}$ is open.

Analyticity of $\operatorname{Per}$ follows from the analyticity of $R$. ∎

Left inverse property of $\operatorname{Per}$

We have

(8)
$\operatorname{Per}\circ\operatorname{Glu}=\operatorname{Id}_{\mathrm{S\text{-}Conf}}.$

Indeed, we defined $\operatorname{Glu}([\cal P])=([f],\operatorname{res})$ for
some, non-unique yet specific, $f\in\cal F$. Using this specific $f$ in the
definition of $\operatorname{Per}$ above immediately yields, from the way $f$