Asymptotic closeness to limiting shapes for expanding embedded plane curves

Abstract

We show that for embedded or convex plane curves expansion, the difference u(x,t)-r(t) in support functions between the expanding curves γ _t and some expanding circles C _t (with radius r(t)) has its asymptotic shape as t→∞. Moreover the isoperimetric difference L²-4πA is decreasing and it converges to a constant $\mathfrak{S} > 0$ if the expansion speed is asymptotically a constant and the initial curve is not a circle. For convex initial curves, if the expansion speed is asymptotically infinite, then L²-4πA decreases to $\mathfrak{S}=0$ and there exists an asymptotic center of expansion for γ _t .