Xuwen Zhu, Stanford University
601 Pao Yue-Kong Library
For a compact Riemann surface M, we would like to understand the deformation theory of constant curvature metrics with prescribed conical singularities. In the positive curvature case, when some or all of the cone angles are bigger than $2\pi$, the analysis is much more complicated than the small angle case. We discover that one key ingredient of the obstructed deformation is related to splitting of cone points. We construct a resolution of the configuration space, and prove a new regularity result that the family of constant curvature conical metrics has a nice compactification as the cone points coalesce, and moreover, the fibrewise family of constant curvature metrics is polyhomogeneous on this compactification. And we hope to apply this new construction to describe the moduli space of spherical conic metrics with no angle constraints. This is joint work in progress with Rafe Mazzeo.