Evolving a convex closed curve to another one via a length-preserving linear flow
Recently S.-T.~Yau proposed the curvature difference flow \[ {{\partial X}\over{\partial t}}(\varphi,t) = \left[ k_1(\varphi,t)-k_2(\varphi)\right] N_1(\varphi,t), \qquad X(\varphi,0)=X_1(\varphi), \qquad \varphi \in {\mathbf S}^1, \leqno(1) \] for closed plane curves \(\gamma_1,\gamma_2\) in \({\mathbb R}^2\) parametrized by smooth embeddings \(X_1,X_2:{\mathbf S^1}\rightarrow{\mathbb R}^2\). Here \(k_1(\varphi,t)\) and \(k_2(\varphi)\) denote the curvatures of the evolving curve \(\gamma_{1,t}\) and of the stationary curve \(\gamma_2\), and \(N_1\) is the outer unit normal vector field to \(\gamma_{1,t}\). Under some appropriate conditions, this flow is expected to evolve \(\gamma_1\) to \(\gamma_2\). Motivated by this and so-called expansion flows, the authors consider the related flow for closed convex curves given by \[ {{\partial X}\over{\partial t}}(\varphi,t) = \left[ {1\over{k_1(\varphi,t)}}-{1\over{k_2(\varphi,t)}} \right] N_1(\varphi,t), \qquad X(\varphi,0)=X_1(\varphi), \qquad \varphi \in {\mathbf S}^1, \leqno(2) \] with notation as before, but where \(k_2(\varphi,t)\) is now the curvature of \(\gamma_2\) at the point at which its outer normal vector is \(N_1(\varphi,t)\). A natural way of studying this flow is to parametrize the two curves by their outer unit normals and to express the flow in terms of the support functions \(u_1\) and \(u_2\) of \(\gamma_1\) and \(\gamma_2\). This leads to the linear evolution problem \[ {{\partial u_1}\over{\partial t}}(\theta,t) = \left[ (u_1)_{\theta\theta}(\theta,t) + u_1(\theta,t) \right] - \left[ (u_2)_{\theta\theta}(\theta,t) + u_2(\theta) \right]. \leqno(3) \] All geometric quantities associated to \(\gamma_1\) can be expressed in terms of \(u_1\). Assuming the condition \(\int_{{\mathbf S}^1} u_1(\theta)\,d\theta = \int_{{\mathbf S}^1} u_2(\theta)\,d\theta\), which implies that the lengths of \(\gamma_1\) and \(\gamma_2\) are equal, the authors prove the following: either the flow (2) exists on some maximal time interval \([0,T_{\text{max}})\) on which \((k_1)_{\text{max}}(t)=\max_{\theta\in{\mathbf S}^1} k_1(\theta,t)\) is unbounded, or the flow exists on \([0,\infty)\) and the evolving closed convex curve \(\gamma_{1,t}\) converges in \(C^\infty\) to \(\gamma_2\) (up to translation) as \(t\rightarrow\infty\). The authors expect that the first possibility of the theorem does not occur, but they are not able to exclude it. They also provide several explicit solutions of the flow to illustrate their results. The proof uses Fourier series techniques. A further result is that if the flow (2) converges in \(C^\infty\) to \(\gamma_2\), then up to a translation this flow is asymptotically the same as the flow given by \[ P(\theta,t)= {1\over{t+1}}P_1(\theta) + \left( 1-{1\over{t+1}}\right)P_2(\theta), \qquad (\theta,t)\in {\mathbf S}^1\times[0,\infty), \] where \(P_1\) and \(P_2\) denote the position vectors of \(\gamma_1\) and \(\gamma_2\).
scientific article; zbMATH DE number 5634874
scientific article