Abstract
We discuss the Funk function \(F(x,y)\) on a Teichmüller space with its Weil–Petersson metric \((\mathcal{T },d)\) introduced in Yamada (Convex bodies in Euclidean and Weil–Petersson geometries, 2011), which was originally studied for an open convex subset in a Euclidean space by Funk [cf. Papadopoulos and Troyanov (Math Proc Cambridge Philos Soc 147:419–437, 2009)]. \(F(x,y)\) is an asymmetric distance and invariant by the action of the mapping class group. Unlike the original one, \(F(x,y)\) is not always convex in \(y\) with \(x\) fixed (Corollary 2.11, Theorem 5.1). For each pseudo-Anosov mapping class \(g\) and a point \(x \in \mathcal{T }\), there exists \(E\) such that for all \(n\not = 0\), \( \log |n| -E \le F(x,g^n.x) \le \log |n|+E\) (Corollary 2.10), while \(F(x,g^n.x)\) is bounded if \(g\) is a Dehn twist (Proposition 2.13). The translation length is defined by \(|g|_F=\inf _{x \in \mathcal{T }}F(x,g.x)\) for a map \(g: \mathcal{T }\rightarrow \mathcal{T }\). If \(g\) is a pseudo-Anosov mapping class, there exists \(Q\) such that for all \(n \not = 0\), \(\log |n| -Q \le |g^n|_F \le \log |n| + Q.\) For sufficiently large \(n\), \(|g^n|_F >0\) and the infimum is achieved. If \(g\) is a Dehn twist, then \(|g^n|_F=0\) for each \(n\) (Theorem 2.16). Some geodesics in \((\mathcal{T },d)\) are geodesics in terms of \(F\) as well. We find a decomposition of \(\mathcal{T }\) by sets, each of which is foliated by those geodesics (Theorem 4.10).
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Notes
After the first draft has been completed, the author was informed that Miyachi-Ohshika-Yamada have been aware of a result similar to Corollary 3.4.
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Acknowledgments
The author is supported in part by a Grant-in-Aid for Scientific Research (No. 23244005). He would like to thank K. Ohshika for discussion. He is grateful to S. Wolpert for comments, and to A. Papadopoulos for his interest in the work. He very much appreciates the comments by the referee. A part of the work was done during visits to the Institut Mittag-Leffler (Djursholm, Sweden) and Max Planck Institute for Mathematics.
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In memory of Professor Shoshichi Kobayashi.
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Fujiwara, K. Geometry of the Funk metric on Weil–Petersson spaces. Math. Z. 274, 647–665 (2013). https://doi.org/10.1007/s00209-012-1089-6
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DOI: https://doi.org/10.1007/s00209-012-1089-6
Keywords
- Funk metric
- Hilbert metric
- Teichmüller space
- Weil–Petersson metric
- Mapping class groups
- Non-symmetric distance