Is every conformal manifold equivalent to a flat one with cone singularities?

At page 10 of

http://page.math.tu-berlin.de/~bobenko/Lehre/Skripte/RS.pdf

it is claimed that the answer is yes.

The answer is yes, and there are several ways to prove it. The result can be restated as "on every Riemann surface there exists a flat conformal metric with conic singularities". In fact the singularities can be prescribed, the only condition is that Gauss Bonnet holds. References: For compact surfaces:

MR1005085 Troyanov, Marc, Prescribing curvature on compact surfaces with conical singularities. Trans. Amer. Math. Soc. 324 (1991), no. 2, 793–821.

For open surfaces:

MR1166122 Hulin, D., Troyanov, M. Prescribing curvature on open surfaces. Math. Ann. 293 (1992), no. 2, 277–315.