November 13, 2020
Curve shortening flow in hyperbolic 2-space.
Geometric flows have emerged as important tools in the application of analysis to geometry. The most basic example of a geometric flow is the curve shortening flow. At its simplest, this is a flow of plane curves, in which a plane curve is deformed in the direction of its normal vector with speed given by the curvature. It has been studied extensively in R2, especially when the flowing curve is closed. In this talk, we study the curve-shortening flow in hyperbolic 2-space. We will prove a general theorem characterizing the curves that evolve in a self-similar manner. Analogous results for curves in the 2-sphere have recently been obtained by Santos dos Reis and Tenenblat (2019) and for curves in the plane by Halldorsson (2012). Time permitting, we will discuss curves that flow so as to remain self-similar up to an overall scaling, including curves that extend to infinity and have a well-defined renormalized length. This is joint work with Ran Xie and should be accessible to undergraduates who have had a first course in curves and surfaces.