Asymptotic behavior of anisotropic curve flows
In this paper, the asymptotic behavior of a closed embedded curve evolving by an anisotropic parabolic equation \(\gamma_t = (\Phi(T)k + \Psi(T))N\) on a Riemannian surface is studied (\(T\), \(N\) are the unit, respectively normal vectors of \(\gamma\), \(k\) denotes its curvature, and the functions \(\Phi\) and \(\Psi\) are rather arbitrary and have to obey some general estimates and restrictions only). Whenever the evolving curve develops a singularity, it shrinks to a point [\textit{J. A. Oaks}, Indiana Univ. Math. J. 43, 959-981 (1994; Zbl 0835.53048)]. The main result in this paper is that the rescaled evolving curve converges to a Minkowski isoperimetrix associated to \(\Phi\) at the shrinking point of \(\gamma_t\) in the \(C^{\infty}\) topology. For the Euclidean case, cf. \textit{K. Chou} and \textit{X. Zhu}, Duke Math. J. 97, 579-619 (1999)].
scientific article; zbMATH DE number 1203421