Consider a polytope with a $2$-dimensional surface and the corresponding metric on this surface (coming from the embedding in $3$-dimensional Euclidean space). Intrinsically the metric is flat everywhere apart from the vertices of the polytope, where one has cone-like singularities if the angle sum does not equal $2\pi$.


Is every conformal manifold equivalent to such a flat cone-manifold? More precisely, is there a sequence of equivalent conformal manifolds that approximates such a singular manifold?



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


