The evolution of area-preserving and length-preserving inverse curvature flows for immersed locally convex closed plane curves

Abstract

In this paper, we study an area-preserving inverse curvature flow and a length-preserving inverse curvature flow for immersed locally convex closed plane curves with rotation number m∈N+. The global-in-time flows are shown to converge smoothly to m-fold round circles as time goes to infinity. The sufficient conditions on initial curve are also found to guarantee the occurrence of the flow's singularity at finite time.