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Holomorphic plumbing coordinates. (English) Zbl 1264.30031
Jiang, Yunping (ed.) et al., Quasiconformal mappings, Riemann surfaces, and Teichmüller spaces. AMS special session in honor of Clifford J. Earle, Syracuse, NY, USA, October 2–3, 2010. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-5340-5/pbk; 978-0-8218-9029-5/ebook). Contemporary Mathematics 575, 41-52 (2012).
This is a report of the current status of the authors’ joint project leading to holomorphic coordinates on augmented Teichmueller space and its quotient under the mapping class group, a compactified moduli space.
Fix a Riemann surface \(R\) of genus \(g \geq 0\) and \(n \geq 0\) punctures satisfying \(2g - 2 + n > 0.\) Using \(R\) as the basepoint, define the Teichmüller space \({T}(R)\) as the space of homotopy classes of quasiconformal mappings. The space \({T}(R)\) is augmented by adding points that represent noded surfaces. Such a surface is a connected complex analytic space containing a finite set of special points called nodes and having the following properties.
i) The complement of the nodes has finitely many connected components, each of which is a finite hyperbolic Riemann surface \(S_{i}\).
ii) Each node has an open neighborhood biholomorphically equivalent to the conical set \[ \big\{(z,w) \in {\mathbb C}^{2} \;:\; z w = 0,\; |z|,|w| < 1\big\} . \]
Let \(X\) and \(Y\) be noded surfaces. A pinch map from \(X\) to \(Y\) is a continuous map with the following properties.
i) If \(q\) is a node in \(Y\), then its inverse image is either a node in \(X\) or a simple closed curve in one of the components \(S_{i}\) of \( X\backslash{q_{j}}(X)\).
ii) The restriction of the map to the inverse image of \(Y\backslash{q_{j}}(Y)\) is an orientation preserving homeomorphism onto \(Y\backslash{q_{j}}(Y)\).
The augmented Teichmüller space \(\widehat{{T}(R)}\) is the set of Teichmüller equivalence classes of these \(R\)-marked surfaces.
Theorem 6.1. The two horocyclic topologies are the same, and they turn \(\widehat{{T}(R)}\) into a Hausdorff space.
Theorem I. Let \(X \in {T}(R)_{P}\) be a \(k\)-noded surface, \(1 \leq k \leq 3g + n - 3\), and \({T}(X)\) its Teichmüller space. Then there exists a biholomorphic map \[ \Phi:{T}(R) \rightarrow U \subset {T}(X)\times(UHP)^{k} \] onto an open subset \(U\), which is surjective onto the first factor \({T}(X)\), and such that every point of \({T}(R)\) has a unique representation as \(Y_{\tau}\) where \(Y \in {T}(X)\) and \(\tau \in (UHP)^{k}\).
If \(X\) is maximally pinched, then \({T}(X) = \{X\}\) and every point of \({T}(R)\) can be uniquely expressed as \(X_{\tau}\) for some \(\tau \in U \subset (UHP)^{3g + n - 3}\).
In this case, the plumbing vectors \(\tau \in U\) serve as global holomorphic coordinates for \({T}(R)\).
For the entire collection see [Zbl 1245.30002].
30F60 Teichmüller theory for Riemann surfaces
32G20 Period matrices, variation of Hodge structure; degenerations