Inventiones mathematicae

, Volume 162, Issue 3, pp 473–492

Asymptotic closeness to limiting shapes for expanding embedded plane curves

Authors

    • Department of MathematicsNational Tsing Hua University
Article

DOI: 10.1007/s00222-005-0449-9

Cite this article as:
Tsai, D. Invent. math. (2005) 162: 473. doi:10.1007/s00222-005-0449-9

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.

Copyright information

© Springer-Verlag 2005