Length-preserving evolution of non-simple symmetric plane curves

Abstract

The length‐preserving nonlocal flow in the plane is investigated for locally convex closed curves, which may be non‐simple. It turns out that for certain classes of symmetric curves, the flows converge to m‐fold circles. Copyright © 2013 John Wiley & Sons, Ltd.

On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves

“…We finally mention that, after Abresch and Langer's celebrated study [1], the behavior of multiply winding rotationally symmetric curves is investigated by several authors for the curve shortening flow [3,20,24,25,44,45] and other second order geometric flows [13,[46][47][48]; in particular, a kind of stability result for multiply covered circles is obtained by Wang [45], in the same spirit of our result. We remark that all these studies focus on locally convex curves, thus using the property of second order.…”

Evolution of Locally Convex Closed Curves in Nonlocal Curvature Flows

“…Later, in [36], two classes of rotationally symmetric and locally convex initial curves, namely, highly symmetric curves and Abresch-Langer type curves (see the definitions in Section 1.2), both enclosing positive algebraic area, are found to guarantee the convergence of the AP flow with α = 1 to m-fold circles. A similar result is established for the LP flow with α = 1 in [37].…”

“…In this paper, we would like to investigate the AP flow and the LP flow for any α > 0 for immersed, locally convex, closed curves. The previous results about singularity formation and convergence when α = 1 (as discussed in [16,36,37]) are generalized to the case when α > 0. Moreover, by observing the sufficient conditions on finite-time singularity or global convergence of the flow, we can compare the difference between the AP flow and the LP flow, and also the difference between the nonlocal flows and the curve shortening flows.…”

On the isoperimetric inequality and surface diffusion flow for multiply winding curves

“…We finally mention that, after Abresch and Langer's celebrated study [1], the behavior of multiply winding rotationally symmetric curves is investigated by several authors for the curve shortening flow [25,24,3,44,45,20] and other second order geometric flows [47,48,13,46]; in particular, a kind of stability result for multiply covered circles is obtained by Wang [45], in the same spirit of our result. We remark that all these studies focus on locally convex curves, thus using the property of second order.…”