# Continuity of the curve-shortening flow with respect to the curve

The curve-shortening flow is an evolution equation for a smooth closed curve $\alpha$ inmersed in a Riemannian surface $M$. The version where $M$ is the Euclidean plane is illustrated for example in this video. The flow is supposed to formalize the idea of evolution by steepest-descent with respect to the length functional on a suitable space of curves.

Letting $\alpha:S^1 \to M$ represent the closed curve (where $S^1$ is the circle) the evolution equation is $$\left\lbrace\begin{array}{l}u(0,x) = \alpha(x)\\ \partial_tu(t,x) = k(t,x)N(t,x)\end{array}\right.$$ where $k$ is the curvature of the curve $u(t,\cdot)$ with respect to the unit normal vector field $N$ (which is chosen to be smooth). Notice that replacing $N$ by $-N$ the curvature would also change sign so that $kN$ remains unchanged.

Short time existence of smooth solutions for this flow was established by Hamilton and Gage using the Nash-Moser implicit function theorem. Other important contributions where made by several people including Grayson who established existence for all positive times and eventual convergence to a closed geodesic in the case of initial curves which are embedded and homotopically non-trivial (the result has a techincal hypothesis: convexity at infinity of $M$).

My question is the following: Does anyone know of a reference for continuity of the solution with respect to the initial curve $\alpha$? That is, how does one establish (or where is it shown) that if $\alpha_n$ is a sequence of embedded simple closed curves converging to a simple closed curve $\alpha$ in the smooth topology then the solutions $u_n$ also converge in the local smooth topology? I'm interested in the case where the solutions $u_n$ are known to exist for all time.

• I would 1st check: Chou, Kai-Seng, and Xi-Ping Zhu. The Curve Shortening Problem. CRC Press, 2010. Unfortunately, I cannot investigate this reference now. My experience is that convergence is robust, but I realize you are seeking more than anecdotal evidence. – Joseph O'Rourke Jan 11 '15 at 0:48
• The curve shortening flow can be written as a nonlinear heat equation, and the continuous dependence on parameters of the solution to the initial value problem for short time follows from a priori estimates for the PDE. – Deane Yang Jan 11 '15 at 1:06