Flat metrics with isolated singularities and polynomial area growth are locally one of three explicit normal forms, and finite-area flat metrics have only conical singularities.
Conformal metrics with constant curvature one and finitely many conical singularities on com pact Riemann surface
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Isolated singularities of flat metrics on Riemann surfaces
Flat metrics with isolated singularities and polynomial area growth are locally one of three explicit normal forms, and finite-area flat metrics have only conical singularities.