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Metrics of constant curvature on a Riemann surface with two corners on the boundary
Jürgen Jost, Guofang Wang and Chunqin Zhou
On a Riemann surface, one of the interesting geometric problems is to determine which functions can be realized as the Gaussian curvature of some pointwise conformal metric. The classical uniformization theorem tell us that every smooth Riemannian metric on a two-dimensional surface is pointwise conformal to one with constant curvature. This question is by now well understood from many different perspectives, and successfully approached by many different methods.
On this basis, research can move on to surfaces with singularities. This, however, is by no means a straightforward generalization of the smooth case. Results for smooth surfaces might not be true for surfaces with singularities. For instance, there exist many surfaces with conical singularities that do not admit a conformal metric of constant Gauss curvature. In fact, a closed surface with two conical singularities admits a conformal metric of constant Gauss curvature if and only if its singularities have the same angle and are in antipodal positions - thus, such a surface necessarily has the shape of an American football; this was proved by Troyanov, Prescribing curvature on compact surfaces with conical singularities, Trans. Amer. Math. Soc. 324(1991), 793-821. Therefore a surface with exactly one singularity (the teardrop) does not carry a conformal metric of constant Gauss curvature.
This result was obtained by methods from complex analysis. It is known, however, that the existence question for conformal metrics is intimately linked to the Liouville equation. In recent years, very powerful PDE methods have been developed to precisely determine the asymptotic behavior of solutions of this equation near singularities.
The purpose of the present paper then is to bring to bear the full force of those methods on the existence problem for conformal metrics with prescribed singularities. In fact, we shall investigate the more general situation of surfaces with boundary. When we have a boundary, the natural curvature condition there, the analogue of the constant Gauss curvature condition in the interior, is the one of constant geodesic curvature.