TITLE: Conformal structure does not see conical singularities
QUESTION [7 upvotes]: the conformal structure does not see the conical singularities of a polyhedral surface.

This is a quote from the Preface of Quantum Triangulations (eds.: Carfora, Marzuoli).
The sentiment is attributed to Marc Troyanov, in a 1991 Trans. AMS paper, 
"Prescribing Curvature on Compact Surfaces with Conical Singularities,"
that I cannot access easily.
Can anyone provide a brief explication or unpacking of the meaning of this intriguing
statement? I'd appreciate it—thanks!

REPLY [4 votes]: Well the question was posted 6 years ago...nevertheless. The reason why the conformal structure does not see the conical singularities of a polyhedral surface is clear if one writes the metric in a neighborhood of a singular point in conformal (isothermal) coordinates. It takes the form  $ds^2 = |z|^{2\beta}|dz|^2$ (if the cone angle is $\theta = 2\pi(\beta+1)$). 
Another argument is that the capacity of a conical singularity is zero, therefore a complex atlas defined from the conformal structure on the complement of the singular set of the polyhedral metric extends to the singular points by Riemann's theorem on removable singularities.