# The hyperbolic metric on a flat surface

Let $S$ be a closed oriented surface of genus $\geq 2$ and $\mathcal{F}_n(S)$ be the space of flat metrics with conic singularities on $S$ whose cone angles are of the form $2k\pi/n$ ($k\in\mathbb{N}$). Let $\mathcal{F}_n^1(S)$ denote metics $g\in\mathcal{F}_n(S)$ of total area $1$.

Each $g\in\mathcal{F}_n(S)$ yields a conformal structure on $S$ and hence a unique hyperbolic metric $g_\mathrm{hyp}$ in the conformal class, which can be written as $$g_\mathrm{hyp}=\lambda_g\cdot g$$ outside the singularities of $g$.

I need some information on the relationship between $g$ and $g_\mathrm{hyp}$, for example,

Question: Fix $n\in\mathbb{N}$. When $g$ runs over $\mathcal{F}_n^1(S)$, does $\int_S\lambda_g^{-1}\mathsf{dvol}_g$ have a upper bound or a positive lower bound?

It seems that when $n=2$, a lower bound exists but upper bounds do not. see the remark below.

It is also natural to ask whether $\max_S(\lambda_g^{-1})$ has upper or lower bounds.

Remark: A metric $g$ belongs to $\mathcal{F}_n(S)$ if and only if it has local expression $$g=|a(z)|^\frac{2}{n}|dz|^2$$ for some conformal structure on $S$ (of which $z$ is a conformal local coordinate) and some holomorphic $n$-differential $\alpha=a(z)dz^n$.

Now assume $n=2$, then the integral in the above question is the Weil-Petersson co-norm of the quadratic differential $\alpha$ (viewed as a cotangent vector to the Teichmüller space). On the other hand, the total area of $g$ is the Finsler co-norm of $\alpha$ with respect to the Teichmüller metric, therefore, for $n=2$, the existence of a lower bound in the above question is equivalent to the majorization of the Weil-Petersson metric by a constant multiple of the Teichmüller metric. The latter majorization is established in a 1974 paper by Linch, but her method is based on Teichmüller geodesics and does not seem to be generalizable to other $n$'s.

Also, I have read that the problem of understanding the relationship between $g$ and $g_\mathrm{hyp}$ is treated by many authors such as Masur, Minsky, Rafi, etc., however, in their papers I only find miscellaneous technical statements on this and can't get a guiding idea. I would appreciate any systematic explanation of that relationship.

• A short proof (just using Cauchy integral formula and collar lemma) of the majoration of the Weil-Petersson by a multiple of the inverse of the systole times the Teichmueller metric is given in Lemma 5.4 (page 38) of this paper here (arxiv.org/pdf/1004.5343.pdf). Maybe this argument can be generalized for your purposes? – Matheus Oct 31 '15 at 15:22
• Thanks for pointing out that nice result! It does generalize to higher-order differentials, but what I need is something stronger, namely, removing the dependence on the inverse of the systole. – Xin Nie Nov 2 '15 at 3:35
• Your argument for the case $n=2$ of your question has a subtle point that I'm missing. In fact, Linch shows that WP distance is at most a scalar multiple of T distance. Of course, this implies a similar estimate for WP and T norms of vectors (Beltrami differentials), but the duality makes that this estimate gets reversed when one tries to use it for covectors (quadratic diff.) as you did (Compare Lemma 5.4 in the paper in my previous comment, where the systole factor "changes" its side depending if we work with vectors/covectors) In particular, the case $n=2$ may have negative answer. – Matheus Nov 3 '15 at 14:46
• Your are right, I overlooked the relation between metrics on a tangent bundle and on a cotangent bundle. I have edited my question. – Xin Nie Nov 4 '15 at 5:51
• I think that the comparison between the WP and T norms proposed right after the statement of Lemma 5.4 in Burns-Masur-Wilkinson paper quoted above (with "systole" replaced by "square root of systole") is essentially sharp near the boundary (Bers regions): indeed, this seems to follow from the asymptotic formulas described in Section 7, page 13, of this survey paper of Wolpert here (arxiv.org/pdf/0801.0175v1.pdf). In particular, this solves your question when $n=2$, do you agree? – Matheus Nov 4 '15 at 12:38

The answer for the case $n=2$ of your question is described in my comments above.
Also, a non-sharp upper bound can be derived from Lemma 5.4 of a paper of Burns-Masur-Wilkinson (quoted above): the advantage of this estimate is that its proof uses only Cauchy formula and collar lemma, and, thus, it can be generalized to any other $n\in\mathbb{N}$.