Evolving convex curves to constant-width ones by a perimeter-preserving flow
The authors study the flow of closed strictly convex smooth plane curves \(\left\{X(\cdot,t)\right\}_{t\in \left[0,T\right[}\) governed by the system \[ \begin{cases} (\frac{\partial X}{\partial t})(\theta,t)=(w(\theta,t)-\eta(\theta,t))N(\theta,t),&(\theta,t)\in S^{1}\times\left]0,T\right[,\\ X(\theta,0)=X_{0}(\theta),&\theta\in S^{1}, \end{cases}\tag{\(\ast\)} \] where \(X(\cdot,t)\) is parameterised by its tangential angle \(\theta\), \(w(\cdot,t)\) is its width and \(\eta(\theta,t)=\rho(\theta,t)+\rho(\theta+\pi,t)\) with \(\rho(\cdot,t)\) its radius of curvature. The main result is that if the pinching condition \(M<3m\) holds, where \(M=\max\limits_{S^{1}} \rho(\cdot,0)\) and \(m=\min\limits_{S^{1}}\rho(\cdot,0)\), then (\(\ast\)) has a unique global solution \(\{X(\cdot,t)\}_{t\in\left[0,\infty\right[}\) of constant perimeter and increasing area with smooth limit curve \(X_{\infty}:S^{1}\rightarrow\mathbb{R}^{2}\) of constant width, and \(X_{\infty}\) is a circle iff \(X_{0}\) is centrosymmetric. The structure of the paper is as follows: Following a motivation of this flow in Section 1, Section 2 is devoted to reducing the study of (\(\ast\)) to that of a linear Cauchy problem for \(\rho\) and \(\eta\). In Section 3 this reformulation is used together with the pinching condition to establish the existence of a unique global solution to (\(\ast\)). Then the area-increasing and perimeter-preserving properties are verified, and finally the asymptotic behaviour of \(X(\cdot,t)\) is studied, culminating in the convergence to a constant-width curve. Here, it is also shown that \(X(\cdot,t)\) becomes more circular in an appropriate sense as \(t\rightarrow\infty\) and that \(X_{\infty}\) is a circle iff \(X_{0}\) is centrosymmetric, which is in strong contrast to related flows such as that of [\textit{M. Gage}, Contemp. Math. 51, 51--62 (1986; Zbl 0608.53002)] where strictly convex curves necessarily converge to circles. Finally, Section 4 is devoted to examples where the pinching condition does not hold, in particular, illustrating numerically that the pinching condition is sufficient but not necessary for the main theorem to hold.
scientific article; zbMATH DE number 6381867