Title 

About Spectral Geometry of Algebraic Curves 

Abstract 

Motivated by algebraic curves, we consider a class of degenerated metrics with finite many cone like singularities on connected compact Riemannian surfaces. The ζ regularized determinant det Δ of the Laplacian can be defined based on the ζ function like in the non singular case. The Polyakov formula, which measures the variation of det Δ under conform variation of the metric, can be adapted to the class of singular metrics we consider. For a special subclass of metrics we establish a uniformization theorem, stating that each of these metrics is conform equivalent to a metric of constant curvature. For other metrics of the considered class there is not such a statement. We give a counterexample. We show that metrics of the subclass mentioned earlier after uniformization are warped products near singularities. It follows immediately that geodesic circles around a singularity are smooth. 

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