A Comparison of Period Coordinates and Teichm\"uller Distance

Let $Q{D}^1({\mathcal{M}}_{g,n})$ be the unit cotangent bundle of the moduli space of Riemann surfaces ${\mathcal{M}}_{g,n}$. There is a metric $d_E$ on $Q{D}^1({\mathcal{M}}_{g,n})$ 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 $(Q{D}^1({\mathcal{M}}_{g,n}),d_E)\to ({\mathcal{M}}_{g,n},d_T)$ is locally a H"older map. We give a lower bound on the exponent in terms of $g$ and $n$.