The geometry of Teichmüller space via geodesic currents
Let S be a compact orientable surface of genus \(g\geq 2\). Denote by \({\mathcal T}(S)\) its Teichmüller space, i.e., the space of isotopy classes of hyperbolic metrics on S. The author provides a very natural compactification of the Teichmüller space \({\mathcal T}(S)\), which concludingly turns out to coincide with W. Thurston's compactification via projective measured laminations [cf. papers of A. Fathi, F. Laudenbach and V. Poenaru in the book ``Travaux de Thurston sur les surfaces'' (1979; Zbl 0406.00016); e.g. \textit{F. Laudenbach}, ibid., 209- 224 (1979; Zbl 0446.57018), \textit{A. Fathi} and \textit{F. Laudenbach}, ibid., 139-150 (1979; Zbl 0446.57015), and \textit{V. Poenaru}, ibid., 5-20 (1979; Zbl 0446.57005)]. The authors construction is based upon the notion of geodesic currents, which has been introduced by himself in an earlier work [cf. the author, Ann. Math. 124, 71-158 (1986)]. The geodesic currents, i.e., the \(\pi_ 1(S)\)-invariant positive measures on the space \(G(\tilde S)\) of (unoriented) geodesics on the universal covering \(\tilde S\) of S, are shown to form a complete uniform space \({\mathcal C}(S)\), whose projectivization \({\mathcal P}{\mathcal C}(S):=({\mathcal C}(S)- 0)/{\mathbb{R}}^+\) is compact. It is then proved that \({\mathcal T}(S)\) admits a proper topological embedding into \({\mathcal C}(S)\), whose image is asymptotic to Thurston's space \({\mathcal M}{\mathcal L}(S)\) of measured laminations. This provides a compactification of \({\mathcal T}(S)\) (within \({\mathcal P}{\mathcal C}(S))\) by \({\mathcal P}{\mathcal M}{\mathcal L}(S)\), so to speak a ``unified'' version of Thurston's approach. Another advantage of the author's construction is that it gives a representation of Teichmüller space \({\mathcal T}(S)\) as a submanifold of an infinite-dimensional analog of the hyperbolic n-space \({\mathbb{H}}^ n\). Then the metric on \({\mathcal T}(S)\) induced by the hyperbolic metric on \({\mathbb{H}}^ n\) is equal (up to a constant factor) to the celebrated Petersson-Weil metric.
scientific article; zbMATH DE number 4065414
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