The evolution of immersed locally convex plane curves driven by anisotropic curvature flow

In this article, we study the evolution of immersed locally convex plane curves driven by anisotropic flow with inner normal velocity V= 1/alpha psi (x)kappa(alpha) for alpha < 0 or alpha > 1, where x is an element of [0, 2m pi] is the tangential angle at the point on evolving curves. For -1 = alpha < 0, we show the flow exists globally and the rescaled flow has a full-time convergence. For alpha < -1 or alpha > 1, we show only type I singularity arises in the flow, and the rescaled flow has subsequential convergence, i.e. for any time sequence, there is a time subsequence along which the rescaled curvature of evolving curves converges to a limit function; furthermore, if the anisotropic function psi and the initial curve both have some symmetric structure, the subsequential convergence could be refined to be full-time convergence.

Automated summary

In this article, the evolution of immersed locally convex plane curves driven by anisotropic flow is studied. The results show that the flow exists globally and converges fully in certain parameter ranges. For other parameter values, only specific types of singularities arise in the flow and the convergence is subsequential.