On Yau's problem of evolving one curve to another: convex case
Let \(X_0\) and \(\tilde{X}\) be two smooth, closed and embedded curves with positive curvature everywhere. In this paper, the authors consider a revised Yau curvature difference flow: \[ \begin{cases} \frac{\partial X}{\partial t}(\varphi,t)=[\kappa(\varphi,t)- \lambda(t)\tilde{\kappa}(\varphi)]N(\varphi,t)\;\text{ in }\;S^1\times(0,\omega),\\ X(\varphi,0)=X_0(\varphi)\;\text{ on }\;S^1, \end{cases} \] where \(\kappa\) and \(\tilde{\kappa}\) are the curvature of the evolving curve \(X\) and the target curve \(\tilde{X}\) separately, \(N\) is the unit inner normal of \(X\), \(\lambda(t)=2\pi/\int_{S^1}\tilde{\kappa}gd\varphi\) and \(g\) is the metric of \(X\). They prove that this flow exists globally on time interval \([0,+\infty)\), preserves the bounded area \(A\) and the convexity of every evolving curve \(X(\cdot,t)\) and deforms \(X_0\) into a fixed curve \(X_{\infty}\) (congruent to \(\sqrt{A/\tilde{A}}\tilde{X}\)) as time tends to infinity, where \(\tilde{A}\) is the area bounded by the curve \(\tilde{X}\).
scientific article; zbMATH DE number 6973891