[Topology and Geometry Seminar] "Neck Pinch Singularities in the Geometry of Surfaces" by Dr. John Loftin
Wednesday, November 14, 2018 - 16:00 to 17:00
Lab 3 B700
The space of all hyperbolic metrics on a closed oriented surface S of genus at least 2 can be compactified by the Deligne-Mumford compactification. It is remarkable that a single type of degeneration, a neck pinch, accounts for all the singular objects adjoined to the moduli space in the Deligne-Mumford limits. In a neck pinch, along a nontrivial simple geodesic loop on S, the hyperbolic length shrinks to zero.
I will generalize this well-known picture to the case of a family of convex RP^2 surfaces which degenerates along a simple loop. A convex RP^2 surface is given as a quotient of a bounded convex domain in RP^2 by a group of projective motions acting discretely and properly discontinuously. Convex RP^2 surfaces generalize hyperbolic surfaces via the Klein model of the hyperbolic plane. I will explain additional geometric phenomena which occur for convex RP^2 structures under neck pinches, and will describe how to put coordinates on the moduli space near the singular surfaces. This is based on joint work with Tengren Zhang.