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

Rafe Mazzeo and Xuwen Zhu

Geometry & Topology 24 (2020) 309–372

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 of these families 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. Of independent interest is how setting up the analysis on these compactified configuration spaces provides a good framework for analyzing “confluent families” of regular singular, ie conic, elliptic differential operators.

Conical metrics, constant curvature, compactified configuration space, polyhomogeneity
Mathematical Subject Classification 2010
Primary: 53C25
Secondary: 58H15
Received: 17 June 2018
Accepted: 24 June 2019
Published: 25 March 2020
Proposed: Gang Tian
Seconded: Simon Donaldson, Tobias H Colding
Rafe Mazzeo
Department of Mathematics
Stanford University
Stanford, CA
United States
Xuwen Zhu
Department of Mathematics
UC Berkeley
Berkeley, CA
United States