Let $S$ be a closed surface of genus $g \geq 2$. Define $\mathrm{Flat}(S)$ to be the set of marked flat metrics on $S$ with cone angles $2\pi+k\pi$ for $k\geq 0$. It is well-known that these all come from holomorphic quadratic differentials, so $\mathrm{Flat}(S)$ is essentially parametrized by the bundle of quadratic differentials over Teichmuller space.
The question is the following. Fix $\epsilon>0$ small and $M>0$ large. Consider $\mathrm{Flat}(S)_{\geq M}$ the set of flat metrics on $S$ with area at least $M$. Is it true that there exists a constant $C>0$ (only dependent of $\epsilon$) such that for every $q \in \mathrm{Flat}_{\geq M}(S)$ the sum of the areas of the balls centered at the zeros of $q$ (i.e. the cone singularities of the metric) and radius $\epsilon$ is bounded by $C$?
I think the answer is yes, and I believe I can prove it if the underlying conformal structure stays in a compact set. I am a bit confused about what can happen in the general case.
Any insight will be very appreciated.
