Chirurgie des surfaces conformes
les mots du titre sont de sa main, en haut à droite du feuillet 1, une
chemise par ailleurs vierge qui ne reçoit pas de numéro de page ; le titre y
est suivi de « (1978 ou 77 ?) », de sa main aussi
Chirurgie des surfaces conformes.
1) Découpage : Découpage d'une surface conforme conforme à bord, avec un
morceau compact de bord compact marqué.
2) Recollage : À partir d'une surface conforme (avec bord
3) Rétrécissage. K \subset X (1-compl. cpct, s. conforme avec
bord)
\widetilde{X} découpé de X suivant K, \partial'\widetilde{X} partie
distinguée du bord, on choisit un multi-anneau autour de
\partial'\widetilde{X}, et alors (un germe de tel anneau autour
d'une composante (\partial'_{i}\widetilde{X})_{i} revient : la donnée d'une
structure riemannienne régulière rectifiable de
long. totale 1 sur celle-ci….) \widetilde{X}^{*} la surface
rétrécie
Choisissons un isomorphisme rectifiable
\partial'\widetilde{X}^{*} \xrightarrow{\ \sim\ } \partial'\widetilde{X}
compatible avec les composantes, et utilisons la relation d'équivalence
de recollement sur \partial'\widetilde{X} pour en déduire une
sur \widetilde{X}^{*}. On en conclut
à gauche, un croquis de \widetilde{X} : deux anneaux hachurés le long
de deux composantes du bord, l'épaisseur de chacun marquée \varepsilon_{i},
et sous le dessin « 0 \leqslant \varepsilon_{i} < +\infty »
4) Élargissage. Opération inverse — au lieu de retrancher une
couronne, on en rajoute une…
5) Opération mixte :
On considère un voisinage tubulaire standard de
\partial'\widetilde{X}, et dans celui-ci, un cercle \Gamma_{i}, et un
isom. rect. \Gamma_{i} \simeq (\partial'\widetilde{X})_{i}. On
C'est une composition rétrécissage \circ élargissage
ou élargissage \circ rétrécissage, au choix
à gauche, un croquis : une composante (\partial'\widetilde{X})_{i}
entourée d'une couronne hachurée, et dans celle-ci le cercle \Gamma_{i}
4') Commutativité de l'opération relativement à diverses
composantes connexes de K.
6) Cas d'une composante de K bordée plongée
\mathbb{R}-analytique isolée. L'opération de revient
donc à une composition de 1°) une couronne intérieure
autour de C, 2°) \mathbb{R}-analytiques
Recoller sur un cylindre les deux bords
joints de la couronne enlevée…
à gauche, trois croquis de cylindres traversés de cercles et de
courbes, le troisième biffé
Relations avec les
7) Modules pour S \subset K \subset X
On suppose compact, et pour simplifier X compact
sans bord (orientée).
Ceci dit, on fixe S_{0} \subset K_{0} \subset X_{0} standard
top., et on rigidifie (S \subset K \subset X) en se
donnant une classe d'isotopie d'homéo. top. avec la situation standard. On
élimine ainsi les automorphismes, sauf dans le cas
Pour avoir un “espace de modules”, on pourrait exiger
v.ca. que les arêtes de K soient des arcs de cercle pour la
connexion projective canonique. Mais les “cartes ”
ne satisfont pas à cette condition ! On pourrait
exiger plutôt que
feuillet de croquis, formules éparses et dessins dans tous les sens ; on
n'en donne que les formules
\widetilde{L}_{0} = \pi^{-1}(L_{0}) \subset \widetilde{K}_{0}
\widetilde{L} = T
\bigl|\widetilde{K}\bigr| - \bigl|\widetilde{L}\bigr| \xrightarrow{\ \sim\ } |K| - |L| ,
\qquad
(\widetilde{K}, \widetilde{L}) \rightsquigarrow
“”
le mot entre guillemets suit la flèche \rightsquigarrow sur la même
ligne
\widetilde{K} \xrightarrow{\ \pi\ } K ,
\qquad
\widetilde{L} = \pi^{-1}(L) \longrightarrow L ,
\qquad
\widetilde{L} \subset \widetilde{K}
\widetilde{K} = \varinjlim_{x \in K \setminus L} K_{x} ,
\qquad
K_{x} = \{\, y \in K \mid y \leqslant x \,\}
\Sigma(X_{i}, \partial X_{i}) = \bigl|\widetilde{K}_{i}\bigr| ,
\qquad \widetilde{K}_{i}
les dessins : une sphère découpée en fuseaux, des arcs emboîtés tracés
dans un disque, une rangée de disques accolés, un cercle muni d'une anse et un
embranchement en Y, c'est-à-dire des « pantalons » ; en marge, des mots
griffonnés illisibles et des coordonnées (x', y')
vaut mieux dès lors exiger que ces arêtes soient
horocycliques par morceaux….
Dans le premier cas (
, en supposant \pi_{0}(K_{0}) \to \pi_{0}(X_{0})
surjectif, des …. nb de modules
réels
N = \Bigl( \sum_{\substack{f \in \pi_{0}(X \setminus K) = F \\ f \ \text{pas un disque}}}
\bigl( 6g(f) + 4\nu(f) - 6 \bigr) \Bigr)
un terme ajouté à droite de la parenthèse est biffé ; en dessous, une
ligne entière est biffée en zigzag, dont on lit seulement « des f qui
des disques »
On suppose X_{0} connexe, K
connexe, \lambda de K
\forall g \geqslant 2 : 3g - 3 où
2 - 2g = \chi = \sum_{f \in F} \bigl( 2 - 2g(f) - \nu(f) \bigr) + 1 - \lambda
i.e.
2g - 2 = \sum_{f \in F} \bigl( 2g(f) - 2 + \nu(f) \bigr) + (\lambda - 1)
g - 1 = \frac{1}{2} \sum_{f \in F} \bigl[ 2g(f) - 2 + \nu(f) \bigr] + \frac{3}{2}(\lambda - 1)
le \tfrac{3}{2} de cette ligne paraît récrit sur un autre chiffre ;
la ligne précédente donnerait \tfrac{1}{2}(\lambda-1)
3(g - 1) = \frac{3}{2} \sum_{f \in F} \bigl[ 2g(f) - 2 + \nu(f) \bigr] + \frac{3}{2}(\lambda - 1)
= \sum_{f \in F} \Bigl[ 3\bigl(g(f) - 1\bigr) + \frac{3}{2}\nu(f) \Bigr] + \frac{3}{2}(\lambda - 1)
(g \geqslant 2)
N - (3g - 3) = \sum_{\substack{f \in F \\ f \ \text{pas un disque}}}
\Bigl( 3g(f) + \frac{5}{2}\nu(f) - 3 \Bigr) + \frac{3}{2}(1 - \lambda)
\ \overset{?}{\geqslant}\ 0
une somme « \sum_{f \in F,\, f \text{ disque}} » devant le dernier
terme est biffée ; dans (1 - \lambda) le signe est récrit
des f qui sont des disques
(g = 1)
N - 1 = \sum_{f \ \text{pas un disque}} \bigl( \underbrace{4\nu(f) - 6}_{\geqslant 2} \bigr) - 1
[NB g(f) = 0 tjs]
brouillon au dos d'une enveloppe ; première des trois versions de la
même liste, reprise aux pages 10 et 12
(1) courbe ell. X + géod. fermée orientée \Gamma
(2) courbe groupe elliptique X_{0} +
sous-tore réel \mathbb{U} + \mathbb{U}-torseur
\Gamma \subset X_{0}(\mathbb{C})
(3) X_{0} + a \in \pi_{1}(X_{0}) primitif + \Gamma
(4)
Torseurs \Gamma sous \mathbb{U}, et Z \subset
\mathbb{C}^{*} discret, Z \simeq \mathbb{Z}
(5)
(6)
sur une enveloppe, un croquis sans légende : un grand disque
contenant un disque plus petit autour d'un point marqué, et deux petits
disques cerclés autour de deux autres points ; à côté, un second contour fermé
avec un point
deuxième version de la liste de la page 8
(1) courbe ellipt. X + géod. fermée orientée \Gamma
mod. tr.
(2) —— X +
(3) —— X + a \in \pi_{1}(X) primitif
(4) Z \subset \mathbb{C}^{*} (\Longleftrightarrow b \in \mathbb{C}^{*},
0 < |b| < 1) s.-gpe discret (\simeq \mathbb{Z})
Z \subset \mathbb{C}^{*} et torseurs \mathbb{T} sous
\mathbb{C}^{*} de Z \subset \mathbb{C}^{*} discret
(5) Droite projective E, (NB
\Delta = E^{*}), et
NB \Delta = E^{*}, T = \mathbb{C}^{*}_{m}, Z = groupe engendré par b)
P = \widehat{E}, \{u, u^{-1}\} = b, b^{-1} \{homothéties
b, b^{-1}\}
(6) Formes G de \mathrm{Gl}(1)_{\mathbb{C}}, +
Borel T \subset B \subset G,
\in B (i.e. G déployé) Couples de
Killing le groupe de \mathrm{Gl}(1)_{\mathbb{C}} ou
torseurs sous T \simeq \mathbb{C}^{*} (i.e. formes
de \mathbb{C}^{*}) [ E']
Z \subset T(\mathbb{C}) = \mathbb{C}^{*} G \supset B \ni b,
½ 2- (i.e. Z b) et |b|
(défini : l'inverse) \neq 1
le 6) est un palimpseste ; la version lisible est celle de la page 12
\text{donc } \{\Gamma\} \simeq X/\Gamma_{0} \simeq X/\mathbb{R}
\simeq \Delta / T_{\mathbb{C}} \cdot Z
\simeq \text{cercles de } P
sous les termes de la chaîne, les numéros cerclés (1), (2), (3'), (4),
(5) des données correspondantes ; un X/\mathbb{Z} est corrigé en
X/\mathbb{R}, et après « cercles de P » viennent, en petit :
« droites projectives P admettant
\mathrm{Fix}(u, u^{-1}) comme bicentre »
troisième version de la liste, au dos d'une enveloppe (voir la page
13) ; les flèches doubles entre les numéros sont les siennes. En tête, une
ligne biffée :
(1) courbe elliptique X + géodésique fermée \Gamma mod. translations
\Updownarrow
(2) courbe elliptique X + géodésique fermée s.-gpe
fermé : i.e. tore réel \Gamma_{0} de la courbe elliptique mod.
translations
\Updownarrow
(3) courbe elliptique X +
\Updownarrow
(4) formes T de \mathbb{G}_{m}, (Z \subset T(\mathbb{C})
(4') formes T de \mathbb{G}_{m},
\in \mathbb{C} et torseurs sous T
complétés \{u, u^{-1}\}
\Updownarrow
(5) droites projectives P, + automorph. u de P non
paraboliques (i.e. ayant 2 pts fixes distincts) et ayant des val. propres
\lambda, \lambda^{-1} telles que |\lambda| \neq 1 CQFS,
:
\Updownarrow
(6) formes G de \mathrm{Gl}(1)_{\mathbb{C}}, + s.-groupe discret
(\simeq \mathbb{Z}) Z \subset G(\mathbb{C})
\{\Gamma\} \simeq \mathrm{or}(\Gamma_{0}) \simeq \text{gén}(Z)
\simeq \text{gén}(Z) \simeq \mathrm{Inv}(T)
sous les termes, les numéros cerclés (1), (2), (3), (3'), (4) ; le mot
devant \{\Gamma\} est illisible (), et « \mathrm{Inv} » est une
lecture douteuse. En marge droite, écrite dans le sens de la hauteur, la
suite de la chaîne : « \simeq \mathrm{Fix}(u) \simeq B_{S} -
(B_{S})^{Z} », avec les numéros (5) et (6)
(3') Variété complexe V de rg 1 + (réseau \Pi) +
(Z \subset \Pi) + torseur X sous V/\Pi
sous Z \subset \Pi, en petit : « facteur direct de rg 1 »
la face de l'enveloppe dont la page 12 est le dos : elle porte le
cachet de l'université de Nagoya daté du 20.12.77. Les calculs qui suivent
y sont écrits tête-bêche par rapport à l'adresse
\{\lambda, \lambda^{-1}\} |\lambda| \neq 1 donc
\lambda \neq \lambda^{-1} i.e. \lambda^{2} \neq 1 i.e.
\lambda \neq \pm 1
\lambda + \lambda^{-1} = \alpha tel que
\alpha \notin [-2, +2] \alpha^{2} - 4
\lambda\bar{\lambda} = 1 \bar{\lambda} = \lambda^{-1}
\lambda + \lambda^{-1} réel
Lettre à Lipman Bers (Les Aumettes, 15.4.1984)
lettre dactylographiée, en anglais, de six pages numérotées à la main
de 1 à 6, en haut au centre (pages 15 à 20), sans signature ; un double
carbone du feuillet 6 suit à la page 21 (lot 2) ; les corrections manuscrites sont intégrées
à leur place et signalées quand elles remplacent un mot tapé. Les lettres
grecques (\nu, \lambda, \rho), le l des longueurs et plusieurs
indices sont portés à la main dans la frappe
Les Aumettes 15.4.1984
Dear Lipman Bers,
Together with Yves Ladegaillerie (a former student of mine) we are running a
microseminar on the Teichmüller spaces and groups, my own motivations coming
mainly from algebraic geometry, and Ladegaillerie's from his interest in the
topology of surfaces. Lately we have met with a problem which I would like to
submit to you, as I understand you are the main expert on Thurston's
hyperbolic geometry approach to Teichmüller space. Before stating the specific
problem on hyperbolic “pants” (which things boil down to), let me tell you
what we are really after.
Assuming given a compact oriented surface with boundary X_{0}
with boundary, giving rise to a modular
space \widetilde{MB}_{g,\nu} (where the letter B recalls that we are
classifying structures with boundary) homeomorphic to
\underline{C}^{d} \times (\underline{R}^{*+})^{\nu}, where the extra factor
(viewed as a component of such a “generalised boundary”). We now should
should get a modular space for “compact conformal oriented surfaces
with generalized boundary” (of type g,\nu and rigidified via X_{0}), call
it \widetilde{MB}_{g,\nu}, homeomorphic to
\underline{C}^{d} \times (\underline{R}^{+})^{\nu}, where now the second
factor corresponds to the “parameters” l_{i}, which are allowed to take also
value 0 (which means that the corresponding component of the generalized
boundary is just one point). Thus \widetilde{MB}_{g,\nu} appears as a
variety with boundary (in the topological sense — in the real analytic sense,
the “boundary” admits “corner-like” points obviously), and
\widetilde{M}_{g,\nu} appears as a part of the boundary.
le + de (\underline{R}^{+})^{\nu} est récrit à la main sur un
autre signe tapé
My interest is in a better geometric understanding of the situation, which
should be “intrinsic” namely not depend on any particular choice of a
surgical decomposition of the reference surface X_{0} into “pants”, used in
order to describe in a handy way the standard “coordinate functions”
on the modular space \widetilde{MB}_{g,\nu}. There appears to be a
geometrically meaningful retraction of \widetilde{MB}_{g,\nu} upon
\widetilde{M}_{g,\nu} (commuting to the operations of the Teichmüller
modular group), the fibers being homeomorphic to
(\underline{R}^{+})^{\nu} — more specifically, I expect the
semi-group (\underline{R}^{+})^{I} (where I is the set of indices for the
“holes” of X_{0}) to act on MB in a natural way, with free action of the
subgroup (R^{+*})^{\nu} upon \widetilde{MB}^{o}, in such a way that
\widetilde{M} is just the quotient of \widetilde{MB} by this action (or of
\widetilde{MB}^{o} by the action of the each fiber F is isomorphic to
(\underline{R}^{+})^{I} by the choice of any “origin” in
F \cap \widetilde{MB}^{o}.
l'exposant de (\underline{R}^{+}) après « homeomorphic to » est
récrit à la main et se lit mal ; « each » tapé est corrigé à la main, et
« F » ajouté
Of course, “computainnally”, in terms of a decomposition of X_{0} into
pants, the idea of such an operation is pretty obvious — namely letting the
components \lambda_{i} of \lambda \in (\underline{R}^{+})^{I} act as a
“multiplier” on the corresponding coordinate l_{i}. However, it is not
clear that this operation is intrinsic — and if it were intrinsic, an
intrinsic geometric description would still be
Of course, in the description of the situation proposed above, the
retraction of \widetilde{MB} upon \widetilde{M} is obtained by
multiplying with the 0 multiplier (all \lambda_{i} are 0). Now there is a
direct geometrical description of this a retraction, by hyperbolic
surgery. Namely, for any compact conformal surface of type g,\nu with
generalized boundary, let's “fill in” the holes which correspond to ordinary
components of the boundary, which are riemanian oriented circles, by “gluing
in” the cones on these circles (which are canonically endowed with a
conformal structure, using the riemanian structure on the given circles).
Thus we get a “functor” from compact conformal surfaces with generalized
boundary (of type g,\nu) to compact conformal surfaces without
boundary, endowed with a system of \nu points (making up a
“wholly degenerate” generalized boundary). When we throw in the
rigidifications and go over to isomorphism classes, this should give the
desired retraction. However, the geometric situation is a lot richer still,
as the compact surface without boundary obtained through surgery is endowed,
not only with a system of \nu points, but moreover with a system of
mutually disjoint discs around these points. The shape of these discs
is by no means arbitrary — we'll say that a system of discs around \nu
points on a compact conformal surface X^{\wedge} without boundary is
“admissible”, if the situation can be obtained as above (up to isomorphism)
from surgery, starting with a compact conformal surface X with boundary.
(NB Among the given “discs”, we should allow that some should be reduced to
their center — we'll call them “degenerate”.) The condition of admissibility
can be expressed intrinsically, by stating that the for every
non-degenerate component \Gamma_{i} of the system of boundaries of those
discs, the two operations we got of the standard circle group (of complex
numbers of module 1) upon \Gamma_{i}, by using the fact that it is (on the
one hand) the boundary of the disc D_{i}, and (on the other hand) that it
is a component of the boundary of the hyperbolic surface
X^{\wedge} \setminus \bigl(\bigcup_{j} D_{j}^{o}\bigr), should be the same.
When X^{\wedge} and the points s_{i} on X^{\wedge} are given, the
possible systems admissible systems of discs around the points
s_{i} depend on \nu parameters — and the first idea which flips to mind
to give a more precise meaning to these “parameters”, is to view them as
being the “radii” of those discs. But then we'll have to define what we mean
by these !
!
The idea here is that, when we have a conformal disc D and an interior
point s of D, then D may be viewed as canonically embedded in the
tangent space T_{s} to D at s, as the “unit disc” at s. Thus, in the
situation above of an defined unique up to
a scalar factor. the set R_{i} of all those possible discs (the
non-degenerate ones, say) may be viewed in a natural way as a “torsor”
(= principal homogeneous space)
« conformal » devant « surface X^{\wedge} », « admissible » devant
« systems » (deux fois), « unique » et l'indice R_{i} sont ajoutés à la
main
under \underline{R}^{+*}, which plays here the role of the parameter space
of all possible (non degenerate) “radii” at s_{i}. If we admit also radius
zero, we accordingly get a parameter space R_{i}^{\wedge}, which may be
viewed as a torsor of sorts over under \underline{R}^{+}. Thus the
set of radii for a given admissible system of discs D_{i} around the points
s_{i} may be viewed as a point of the product-space
r = (r_{i})_{i\in I} \in R^{\wedge} = \prod_{i\in I} R_{i}^{\wedge} .
My expectation is that an admissible set of discs (D_{i}) is well
determined by the knowledge of the corresponding set of radii r, and
moreover that a given set of radii r corresponds to an admissible system of
admissible discs iff it satisfies a set of unequalities
r_{i} < \rho_{i} ,
where
\rho = (\rho_{i})_{i\in I} \in R = \prod_{i\in I} R_{i}
is a some fixed system of radii, corresponding to a fixed system of
choices of hermitian metrics in the tangent spaces T_{s_{i}}.
I now see that this “expectation” does'nt quite match with the previous one,
about a “natural operation” of (\underline{R}^{+})^{I} upon \widetilde{MB},
having certain properties — it would match only if all \rho_{i} where equal
to +\infty (hence not in R_{i} itself strictly speaking). I must confess I
did'nt look too thoroughly yet at the situation, and moreover I've been busy
with rather different kind of things for the last two or three months, and
lost a little contact contact. What is clear however is that the
main key to an understanding of the general situation, is in an understanding
of the basic particular case of Thurston's pants. If we number 0, 1, \infty
the three “holes” of such a pant, the surface X^{\wedge} can be identified
canonically to the Riemann sphere, and the basic question then is to
understand how the pant is imbedded in this sphere \Sigma, as a complement
of (open) the union of (open) discs around the points
0, 1, \infty, these discs forming an “admissible system”. So the main
question is about understanding the structure of all possible admissible
systems of three discs on \Sigma.
Puzzling a little about this problem, the following model came to my mind
(corresponding to “limiting radii” \rho_{i} which are finite, not
infinite). I view \Sigma as endowed with it's usual euclidean metric, for
which the real projective line is a great circle, with 0, 1, \infty at
equal distance from each other on this equator. These points may be viewed
as the centers of three “orange quarters slices”, making up a
cellular subdivision of \Sigma, where the common boundary of two among the
“quarters slices” Q_{i} (i \in \{0, 1, \infty\}) is a
half-great circle passing in between s_{i} and s_{j} at equal
« all », « r », « an admissible », « some », « with », « contact »
et les deux « slices » sont de sa main, en interligne ; dans
(\underline{R}^{+})^{I}, l'exposant + est récrit sur un signe tapé
distance from both, these three half-circles joining at the two poles P^{+}
and P^{-}. The “disc” Q_{i} around s_{i} has a conical structure around
s_{i} (as has any conformal pointed disc), and we may take the concentric
discs discs \lambda_{i} Q_{i} with
0 < \lambda_{i} < 1 .
The model I had in mind was that the (non degenerate) admissible systems of
discs around the points s_{i} (i \in \{0, 1, \infty\})
This model, if correct, would give a rather precise description of the
inclusion relationships between pants, when these are considered as embedded
in the sphere. The intersection of all would be this system of three half
circles C_{i}, and the two poles P^{+}, P^{-} would play a significant
role in the geometry of the pants, from this point of view. But it does'nt
seem that neither those half circles (which need not be geodesical I guess),
nor the two poles, have ever been described as intrinsically associated to a
pant. Of course, this model would give alternative “parameters” \lambda_{i}
for describing a pant, which are best suited for grasping the pants in terms
of spherical geometry. The next question would be an understanding of the
relationship between these parameters, and Thurston's l_{i}. Maybe it is
unreasonable to expect that for given index i \in \{0, 1, \infty\}, the
length l_{i} depends only on \lambda_{i} and not on the other parameters
\lambda_{j} — and for this reason,
My question of course is whether you have any information or idea to
propose, especially on the basic problem of relying pants to spherical
geometry, and more specifically, whether the model above is likely to hold,
or is definitely false. Also, one difficulty we found with hyperbolic
geometry of conformal surfaces, is that apart from existence and unicity of
the hyperbolic structure (compatible with the given conformal one and for
which the boundary is geodesic), there seems to be little
hold on more specific properties. As an example, starting with a compact
conformal surface with boundary X (a pant, say), of hyperbolic type, and
removing an (open) “collar” around the boundary, we get another surface with
boundary X' — what about the relation of between the two
corresponding metrics ? Assuming the model for pants above is correct, it
would be nice to have an explicit expression of the metric of a pant in terms
of the parameters \lambda_{i}.
With my thanks for your attention, and for whatever comment you will care to
make, very sincerely ypur's
« compact » et « between » sont ajoutés à la main ; la lettre s'arrête
sur la formule de politesse, sans signature sur ce feuillet