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We show that any initial closed curve suitably close to a circle flows under length-constrained curve diffusion to a round circle in infinite time with exponential convergence. We provide an estimate on the total length of time for which such curves are not strictly convex. We further show that there are no closed translating solutions to the flow and that the only closed rotators are circles.
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A recent article by the first two authors together with B Andrews and V-M Wheeler considered the so-called `ideal curve flow, a sixth order curvature flow that seeks to deform closed planar curves to curves with least variation of total geodesic curvature in the $L^2$ sense. Critical in the analysis there was a length bound on the evolving curves. It is natural to suspect therefore that the length-constrained ideal curve flow should permit a more straightforward analysis, at least in the case of small initial `energy. In this article we show this is indeed the case, with suitable initial data providing a flow that exists for all time and converges smoothly and exponentially to a multiply-covered round circle of the same length and winding number as the initial curve.
In this paper, we consider a new length preserving curve flow for convex curves in the plane. We show that the global flow exists, the area of the region bounded by the evolving curve is increasing, and the evolving curve converges to the circle in C-infinity topology as t goes to infinity.
In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Gigas conjecture) and the shape of Type I singularities (Chous conjecture).
It is proved the generalization of Toponogov theorem about the length of the curve in two-dimensional Riemannian manifolds in the case of two-dimensional Alexandrov spaces.
We survey techniques for constrained curve fitting, based upon Bayesian statistics, that offer significant advantages over conventional techniques used by lattice field theorists.
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