# Geometry of the Funk metric on Weil–Petersson spaces

Research paper by Koji Fujiwara

Indexed on: 08 Nov '12Published on: 08 Nov '12Published in: Mathematische Zeitschrift

#### 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).