A Comparison of Period Coordinates and Teichm\"uller Distance

Abstract

Let $QD^1(\mathcal{M}_{g,n})$ be the unit cotangent bundle of the moduli space of Riemann surfaces $\mathcal{M}_{g,n}$. There is a metric $d_E$ on $QD^1(\mathcal{M}_{g,n})$ that is locally bi-Lipschitz to the Euclidean metrics defined by systems of period coordinates coming from of short and moderate-length saddle connections. We show the following: if $\mathcal{M}_{g,n}$ is equipped with the Teichm\"uller metric $d_T$, then the projection $(QD^1(\mathcal{M}_{g,n}),d_E) \to (\mathcal{M}_{g,n},d_T)$ is locally a H\"older map. We give a lower bound on the exponent in terms of $g$ and $n$.