On the formation of singularities in the curve shortening flow (Q807983)

Let X: \(S^ 1\times [0,T)\to {\mathbb{R}}^ 2\) be a family of convex immersed plane curves satisfying the ``curve shortening equation'' \(\partial X/\partial t=kN\) where k is the curvature and N the unit normal of the curve X(.,t). Assume that T is maximal, i.e. the curves X(.,T) converge to a singular curve for \(t\to T\). Let \(\kappa(t)\) be the maximal curvature of the curve X(.,t). It is easy to see that \(\kappa(t)\) blows up at least like \((T-t)^{-1/2}\) for \(t\to T.\) The results of the paper are the following. Theorem A. If \(\kappa(t)\) does not blow up faster than \((T-t)^{-1/2}\) then the curves X(.,t) shrink to a point in an asymptotically self-similar manner. Theorem B. For the blow up of the curvature one has the following rough upper bound: \(\lim_{t\to T}(T-t)\kappa (t)=0.\) Theorem C. If \(\kappa(t)\) blows up faster than \((T-t)^{-1/2}\) then there is a sequence \(t_ n\to T\) such that the curve obtained by magnifying \(X(.,t_ n)\) so that its maximal curvature becomes 1 will converge to the graph of \(y=-\log\cos x\). Theorem D. If the total curvature which disappears into the singularity is less than \(2\pi\), then it must actually be \(\pi\) and the statement of Theorem C holds for any sequence \(t_ n\to T\). Furthermore, for any \(\epsilon>0\) there is a constant \(C_{\epsilon}\) such that \(\kappa (t)\leq C_{\epsilon}/(T-t)^{+\epsilon}\).