Shortening space curves and flow through singularities (Q1203612)

When a closed curve immersed in the plane evolves by its curvature vector, singularities can form before the curve shrinks to a point. Calabi suggested a method for flowing through planar singularities using space curves. The idea is to take a family \(\Gamma\) of embedded space curves limiting on the immersed plane curve, and then define a flow through the singularity as the limit of the flows in \(\Gamma\). To realize this procedure, the author defines a ramp as a space curve which steadily gains height, that is, its tangent vector has positive vertical component at all points. Then, let \(\Gamma_ 0\) be an evolving immersed closed curve in the plane, non-singular for \(t\in [0,\omega_ 0]\). Let \(\Gamma_{\lambda}(0)\), \(\lambda\in (0,1]\), be any family of periodic ramps which project vertically into \(\Gamma_ 0(0)\) and have vertical period \(\lambda\). Let \(\Gamma_{\lambda}(t)\) be the corresponding evolution by the curvature flow for space curves. The author shows that Calabi's method works in this case by proving the following theorem: Given \(\Gamma\) as above, we have (1) that \(\Gamma_{\lambda}\) exists for all \(\lambda>0\) and is smooth for all time, (2) the limit as \(\lambda\to 0\) of \(\Gamma_{\lambda}(t)\) is a smooth curve for all but a finite number of times \(\omega_ i\in [\omega_ 0,\omega_ n]\), being a point for \(t\geq\omega_ n\), (3) the limit agrees with the planar evolution away from the singularities, (4) the limit planar curve is independent of the choice of \(\Gamma_{\lambda}(0)\) and hence is unique.