Curve shortening flow
Curve shortening flow is the gradient flow of length for regular curves. It drives an immersed curve on a Riemannian surface (e.g. the plane) with velocity equal to its curvature vector. Appropriately interpreted, this (differentio-geometric) process may be studied from the point of view of parabolic partial differential equations.
Curve shortening flow models the evolution of grain boundaries and the shapes of worn stones in two dimensions, and has been exploited in a multitude of further applications.
The curve shortening flow has much better convergence properties than other geometric flows. Indeed, Grayson (1987), building on work of Gage–Hamilton (1986), proved that it "untangles" every closed embedded planar curve, eventually causing the curve to contract a single point, with limiting shape that of a round circle. In fact, this behaviour holds in more general ambient spaces (e.g. compact surfaces), leading to numerous geometric applications (e.g. simple proofs of the Lusternik–Schnirelmann theorem on the existence of at least three closed geodesics on any Riemannian two-sphere and Smale’s theorem on the retractibility of Diff(S2) to SO(3)). The curve shortening flow also improves the isoperimetric ratio of an embedded curve, leading to further applications.
The natural Neumann boundary value problem for curve shortening flow, called the free boundary problem, asks for a family of curves whose endpoints lie on (but are free to move on) a fixed barrier curve which is met by the solution curve orthogonally. In this setting, it is known that convex curves with free boundary on a smooth, convex, locally uniformly convex barrier remain convex and shrink to a point on the barrier curve with limiting shape that of a round semicircle.
Despite its relative simplicity and the intensity with which it has been studied since the 1980's, many interesting questions about curve shortening flow (and other closely related flows) remain open.
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