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Degenerations of quadratic differentials on \(\mathbb {CP}^1\). (English) Zbl 1146.30020

The article deals with families of flat metric on surfaces of genus zero, where the flat metrics are assumed to have conical singularities, \(\mathbb Z/2\mathbb Z\) linear holonomy and a fixed vertical direction. The moduli space of such metrics is isomorphic to the moduli space of meromorphic quadratic differentials on \(\mathbb C\mathbb P^1\) with at most simple poles and is naturally stratified by the number of poles and by the orders of zeros of a quadratic differential. Any stratum is non compact and a neighborhood of its boundary consists of flat surfaces that admit saddle connections of small length. The structure of the neighborhood of the boundary is also related to counting problems in a generic surface of the stratum (the “Siegel-Veech constants”). When the length of a saddle connection tends to zero, some other saddle connections might also be forced to shrink. In the case of an abelian differential this corresponds to homologous saddle connections. In the general case of quadratic differentials, the corresponding collections of saddle connections on a flat surface are said to be ĥomologous. (pronounced “hat-homologous”). Usually, the study of the structure of the neighborhood of the boundary is restricted to a thick part, where all short saddle connections are pairwise ĥomologous. Following this idea, the author considers the complement of the codimension 1 subset \(\Delta\) of flat surfaces that admit a pair of saddle connections that are both of minimal length, but which are not ĥomologous. For a flat surface in the complement of \(\Delta\), it is defined the configuration of the maximal collection of homologous saddle connections that contains the smallest saddle connection of the surface. This defines a locally constant map outside \(\Delta\).
The main result of the paper is the following: Let \(\mathcal Q_1(k_1, \dots , k_r)\) be a stratum of quadratic differentials on \(\mathbb C\mathbb P^1\) with at most simple poles. There is a natural bijection between the configurations of ĥomologous saddle connections existing in that stratum and the connected components of \(\mathcal Q_1(k_1,\dots,k_r)\backslash\Delta\). The connected components of \(\mathcal Q_1(k_1,\dots,k_r)\backslash\Delta\) are called the configuration domains of the stratum.
These configuration domains might be interesting to the extend that they are “almost” manifolds in the sense of the following corollary. Let \(\mathcal D\) be a configuration domain of a stratum of quadratic differentials on \(\mathbb C\mathbb P^1\). If \(\mathcal D\) admits orbifoldic points, then the corresponding configuration is “symmetric” and the locus of such orbifoldic points are unions of copies (or coverings) of submanifolds of smaller strata. Restricting ourselves to the neighborhood of the boundary, the following propositions are shown: Let \(\mathcal D\) be a configuration domain of a stratum of quadratic differentials on \(\mathbb C\mathbb P^1\). Then \(\mathcal D\) has only one topological end. Any stratum of quadratic differentials on \(\mathbb C\mathbb P^1\) has only one topological end.

MSC:

30F30 Differentials on Riemann surfaces
57R30 Foliations in differential topology; geometric theory
32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables)
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References:

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