, Volume 162, Issue 3, pp 473-492

Asymptotic closeness to limiting shapes for expanding embedded plane curves

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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 L 2-4πA 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 L 2-4πA decreases to \(\mathfrak{S}=0\) and there exists an asymptotic center of expansion for γ t .

Mathematics Subject Classification (2000)

35K15, 35K55