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,



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> > **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?
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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](https://en.wikipedia.org/wiki/Weil%E2%80%93Petersson_metric) 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](https://en.wikipedia.org/wiki/Teichm%C3%BCller_space#Teichm.C3.BCller_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.



The hyperbolic metric on a flat surface
The answer for the case $n=2$ of your question is described in my comments above. 


In summary, the lower bound can be derived from Linch's paper on the comparison of Teichmuller and Weil-Petersson distances, while an essentially sharp upper bound involving the inverse of the square root of the systole function can be derived from Wolpert's asymptotic formulas in Bers regions. 


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 $n\in\mathbb{N}$. 


