An application of interpolation inequalities between the deviation of curvature and the isoperimetric ratio to the length-preserving flow

TL;DR

This paper applies new interpolation inequalities relating curvature deviation and isoperimetric ratio to analyze the long-term behavior of length-preserving flows of closed plane curves without assuming convexity.

Contribution

It introduces novel interpolation inequalities and uses them to study the asymptotic behavior of non-convex length-preserving curve flows.

Findings

Established large-time existence and convergence properties

Extended analysis to non-convex curves

Connected curvature deviation with isoperimetric ratio dynamics

Abstract

In recent work of Nagasawa and the author, new interpolation inequalities between the deviation of curvature and the isoperimetric ratio were proved. In this paper, we apply such estimates to investigate the large-time behavior of the length-preserving flow of closed plane curves without a convexity assumption.