Inventiones mathematicae

, Volume 162, Issue 3, pp 473–492

Asymptotic closeness to limiting shapes for expanding embedded plane curves

Article

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 γt and some expanding circles Ct (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 γt.

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Copyright information

© Springer-Verlag 2005

Authors and Affiliations

  1. 1.Department of MathematicsNational Tsing Hua UniversityHsinchuTaiwan

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