#### Vol. 290, No. 2, 2017

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Length-preserving evolution of immersed closed curves and the isoperimetric inequality

### Xiao-Liu Wang, Hui-Ling Li and Xiao-Li Chao

Vol. 290 (2017), No. 2, 467–479
##### Abstract

It is shown that all immersed closed, locally convex curves with total curvature of $2m\pi$ and $n$-fold rotational symmetry ($m∕n\le 1$) finally evolve into $m$-fold circles under the length-preserving curvature flow. Sufficient conditions for the occurrence of the finite-time singularities in the flow are also established. As a byproduct, an isoperimetric inequality for rotationally symmetric, locally convex curves is proved via the flow method.

##### Keywords
curvature flow, nonlocal, blow-up, convergence, isoperimetric inequality
##### Mathematical Subject Classification 2010
Primary: 35B40, 35K59, 53C44