# Asymptotic closeness to limiting shapes for expanding embedded plane curves

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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

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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}-4π*A* 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}-4π*A* decreases to \(\mathfrak{S}=0\) and there exists an asymptotic center of expansion for γ_{t}.