Conical metrics on Riemann surfaces, I: the compactified configuration space and regularity

Abstract

We introduce a compactification of the space of simple positive divisors on a Riemann surface, as well as a compactification of the universal family of punctured surfaces above this space. These are real manifolds with corners. We then study the space of constant curvature metrics on this Riemann surface with prescribed conical singularities at these divisors. Our interest here is in the local deformation for these metrics, and in particular the behavior as conic points coalesce. We prove a sharp regularity theorem for this phenomenon in the regime where these metrics are known to exist. This setting will be used in a subsequent paper to study the space of spherical conic metrics with large cone angles, where the existence theory is still incomplete.

68 pages, 6 figures

References in corpus (2)

• Drawing cone spherical metrics via Strebel differentials
• Spherical metrics with conical singularities on 2-spheres

Cited by in corpus (6)

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• Spectral properties of reducible conical metrics
• Prescribing Gaussian curvature on surfaces with conical singularities and geodesic boundary
• Irreducible cone spherical metrics and stable extensions of two line bundles
• Cone spherical metrics and stable vector bundles