Geometry of the Funk metric on Weil–Petersson spaces

Abstract

We discuss the Funk function \(F(x,y)\) on a Teichmuller 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).