The Isolated Singularities Of Conformal Flat Metrics

This paper devotes to give a comprehensive characterization of the singularities of conformal flat metrics on Riemann surfaces.It can be regarded as a generalization of proposition 4 of[8],in which Bryant dealt with the elliptic case and concluded that the singularities of conformal metrics of Gauss curvature one and finite area must be conical singularities.It comes naturally to ask how about the flat ones.We consider a metric of curvature zero and the area of which grows by polynomial near the singularities.Different from the elliptic case,the behavior of the singularities at this moment is no longer just like the conical ones,but appears some other types,which is astonishing.