강현석 (GIST)

Asymptotic behavior of convex curve shortening flows

 이태훈 (KIAS)

Asymptotic behavior of complete graphical curves under the curve shortening flow

     The curve shortening flow describes the motion of a planar curve that evolves by its curvature vector. For closed curves, the behavior of the flow is relatively simple, i.e., the curve shrinks to a point in finite time. On the other hand, for some complete non-compact curves (specifically, curves that divide the plane into two domains with infinite areas), the flow exists for all time. In particular, curves given by a graph of a function fall into the latter case so that the evolution of graphs exists for a long time. 

     This mini-course aims to address the long time behavior of curves given by graphs under the curve shortening flow. For initial data, we focus on curves that have bounded gradients with an additional assumption on the behavior at infinity. We shall prove that such initial curves converge to a solution of the textit{expanding self-similar equation} after appropriate rescaling.

     The first two hours will be devoted to deriving evolution equations for basic quantities such as gradient function, curvature, and so on. In the last hour we will discuss the convergence of the rescaled curves.

최범준 (POSTECH)

SHARP CONVERGENCE RATE OF CURVE SHORTENING FLOW