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Geometric Analysis and Partial Differential Equations Seminar
September 18, 2014 @ 10:30 am - 11:30 pm
Scott Parkins (University of Wollongong)
The Generalised Polyharmonic Curve Flow of Closed Planar Curves
We consider a family of higher-order curvature flows on closed planar curves (which includes the curve shortening flow and curve diffusion flow). We look at a natural energy called the “normalised oscillation of curvature” that measures how far a closed curve is from being an embedded circle (in an averaged $L^2$ sense not unlike the Willmore energy for closed surfaces). We then show that under any of these flows, closed curves suitably close to a circle (i.e. with a small normalised oscillation of curvature, as well as satisfying a suitable isoperimetric condition) exist for all time and converge exponentially fast to a round embedded circle.