# Reference request: flat surfaces

When writing a paper, I feel like to point out exact references to the following seemly easy facts concerning flat structures on a closed surface $\Sigma$ with negative Euler characteristic:

1. The universal cover $\widetilde{\Sigma}$ equipped with the pullback of any flat metric with conic singularity is quasi-isometric to the hyperbolic plane.
2. Define two flat metrics $g_1$ and $g_2$ by means of holomorphic $n$-differentials $U_1$ and $U_2$. Then the pullbacks of $g_1$ and $g_2$ to $\widetilde{\Sigma}$ are quasi-isometric and the ratio of quasi-isometry can be controlled by $\|U_1-U_2\|$.

Please either give me a reference or a simple proof/explanation. Any help is appreciated!

• (1) is probably not what you mean, since all compact metric spaces are quasi-isometric to each other. Do you mean the pullback metric on the universal cover? or do you mean quasi-conformal instead of quasi-isometric? – John Pardon Mar 13 '15 at 20:11
• You are right, I actually work on the universal cover. I've edited the question. – Xin Nie Mar 13 '15 at 20:34