  Robert Bryant (Theorie des varietes minimales et applications, 1988, 154:
321-347) proved that an isolated singularity of a conformal metric of positive
constant curvature on a Riemann surface is a conical one. Using Complex
Analysis, we find all of the local models for an isolated singularity of a flat
metric whose area satisfies some polynomial growth condition near the
singularity. In particular, we show that an isolated singularity of a flat
metric with finite area is also a conical one.
