Cote n° 67 · batch 1 · pages 1–20
· Transcription · Chirurgie des surfaces conformes : notes manuscrites (s.d., 1983-1984), lettres (1984)
Datation de l’inventaire : [à partir de 1977]-1984
Édition de démonstration
TEI P5 source — open the XML · download batch-01.fr.xml
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
2Chirurgie des surfaces conformes.
1) Découpage : Découpage d'une surface conforme […] (évent. avec bord) suivant un sous-1-complexe rectifiable compact : On trouve une surface conforme à bord, avec un morceau compact de bord compact marqué.
2) Recollage : À partir d'une surface conforme (avec bord […]) et d'un morceau compact du bord, et une relation d'équivalence “admissible” (au sens rectifiable) sur le bord, […] on trouve 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 […] à bord […] déduite […] la couronne.
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 […] \(X^{*}\) et \(K \subset X^{*}\).
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 « \(0 \leqslant \varepsilon_{i} < +\infty\) »
34) Élargissage. Opération inverse — au lieu de retrancher une couronne, on en rajoute une…
5) Opération mixte : […] le nom de l'opération, souligné, commence par un p et finit en -age ; il revient au 4') et au 6) ci-dessous et n'a pu être lu
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 […] définit […] considère \(\widetilde{X}^{*}\) déduit de \(\widetilde{X}\) en enlevant \(K_{i}\) “la […] partie de \(T_{i}\) entourée par \(\Gamma_{i}\)”. On termine comme dans 3°), 4°) pour former \(\widetilde{X}^{*}\) et \(K \subset X^{*}\).
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 […] […] plongés en forme d'une petite couronne…
47) 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 […] où \(X_{0}\) a des composantes de genre \(0, 1\) ….
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 […] […] les paramétrisations géodésiques sur le disque, […] disque avec l'[…] hyperbolique […] de vecteurs aux […]. [Éventuellement on pourrait exiger l'équidistance des sommets […] des bords de chaque composante […] le disque…] Mais on a des choix de disques […] par les morceaux […] : pas de […] constant. Il
5feuillet 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 \((x', y')\)
6vaut mieux dès lors exiger que ces arêtes soient horocycliques par morceaux….
Dans le premier cas ([…] exigé que les disques ou le disque […] […]…)
[…], 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]
8brouillon 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)
9sur 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
10deuxième version de la liste de la page 8
(1) courbe ellipt. \(X\) + géod. fermée orientée \(\Gamma\) mod. tr.
(2) —— \(X\) + […] […] \(\mathbb{U}\) (tore compact standard) \(\subset X_{0}\)
(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 […] \(b \in \mathbb{C}\), \(0 < |b| < 1\)
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 »
12troisième 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 \(X\) + […] sous-groupe facteur direct libre de rg 1 de \(\pi_{1}(X)\)
\(\Updownarrow\)
(4) formes \(T\) de \(\mathbb{G}_{m}\), (\(Z \subset T(\mathbb{C})\) […], et torseurs sous \(T\) \(b \in \mathbb{C}\), \(0 < |b| < 1\)
(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, […] : […] \(\lambda\) […] pour \(|\lambda| > 1\))
\(\Updownarrow\)
(6) formes \(G\) de \(\mathrm{Gl}(1)_{\mathbb{C}}\), + s.-groupe discret \((\simeq \mathbb{Z})\) \(Z \subset G(\mathbb{C})\) […] non unipotent (i.e. formé d'élts semi-simples, i.e. contenu dans un tore maximal \(T\))
\[\{\Gamma\} \simeq \mathrm{or}(\Gamma_{0}) \simeq \text{gén}(Z) \simeq \text{gén}(Z) \simeq \mathrm{Inv}(T)\] 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 »
13la 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
15Les 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 \(X_{0}\) […] as a reference-surface for constructing the Teichmüller-type spaces, of genus \(g\) and with “holes” (satisfying \(2g-2+\nu > 0\)), my primary interest is in the more “algebraic” version of Teichmüller space, corresponding to the question of classifying algebraic non singular curves over \(\underline{C}\), of genus \(g\), with a system of [\(\nu\)] points (all distinct) given on \(X\), together with a “Teichmüller rigidification” of \((X,S)\) namely a homotopy equivalence between \(X_{0}\) and \(X \setminus S\). I'll denote this space, homeomorphic to \(\underline{C}^{d}\) (where \(d = 3g-3+\nu\)), by \(\widetilde{M}_{g,\nu}\) (the tilda suggesting that it is the universal covering of a finer object I am still more interested in, namely the algebraic variety (or rather “multiplicity”, or “stack” in the terminology of Mumford-Deligne) of moduli for algebraic curves of type \((g,\nu)\)). Thurston however considers a different modular space, where algebraic curves with a given system of points are replaced by compact conformal oriented surfaces 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 […] corresponds to the extra parameters introducing through the existence of the boundary, namely the length's of the […] components of the boundary with respect to the canonical hyperbolic structure on the given surface. Our interest is in pinpointing the precise relationship between the two modular spaces. The obvious idea here is to consider the case of an algebraic curve with \(\nu\) points given as a limit-case of a compact conformal surface with boundary, when all the lengths \(l_{i}\) of the components of the boundary tend to zero. Therefore, it looks suitable to consider both modular spaces above as imbedded in a larger third one, which corresponds to the same modular problem as in Thurston's theory, except that we allow the “boundary” to have some components reduced to just one point, in the neighbourhood of which \(X\) is just a conformal surface without boundary, but with a given point la frappe laisse en blanc le nombre de points après « a system of » ; « compact », devant « conformal surface with boundary », est ajouté à la main
16(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 […] corresponding subgroup), and that 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 […] desired.
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
17“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 […] admissible system of discs \((D_{i})_{i\in I}\) around \((s_{i})_{i\in I}\), for every \(s_{i}\) corresponding to a non-degenerate \(D_{i}\), we get a canonical disc \[\Delta_{i} \subset T_{s_{i}}\] in the tangent space — and of course, for degenerate \(D_{i}\), we'll take \(\Delta_{i}\) to be degenerate too. The discs we get in a given \(T_{s_{i}}\) (for a fixed system \((s_{i})\), and a variable admissible system of discs around these \(s_{i}\)) are all discs in the strict euclidean sense, given by an unequality \[|z| \leqslant r_{i} ,\] where \(z \mapsto |z|\) denotes some hermitian metric on \(T_{s_{i}}\) compatible with the conformal structure — this metric being 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
18under \(\underline{R}^{+*}\), 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
19distance 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 \(s_{i}\) (\(i \in \{0, 1, \infty\}\)) […] are exactly the systems of discs \(\lambda_{i} Q_{i}\), with \(\lambda_{i}\) as above. (If we allow some discs to be degenerate, this means that instead of the unequality above we merely demand \(0 \leqslant \lambda_{i} < 1\), 0 not excluded.)
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, […] the intuition at the beginning of this letter, using Thurston's coordinate functions and notably the \(l_{i}\)'s to get a fibration structure on \(\widetilde{MB}\) over \(\widetilde{M}\), in terms of a given decomposition of \(X_{0}\) into pants, is probably not really relevant, namely it is non intrinsic. Assuming the model I am suggesting is correct, the accurate description […] of \(\widetilde{MB}\) in terms of \(\widetilde{M}\) would be \[\widetilde{MB} \simeq \widetilde{M} \times [0, 1[^{I} ,\] where the second factor on the right hand sight refers to the system of multipliers \(\lambda_{i}\) (\(i \in I\)), tied to the \(r_{i}\) above by \(r_{i} = \lambda_{i} \rho_{i}\).
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
20hold 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