Interpolation inequalities between the deviation of curvature and the isoperimetric ratio with applications to geometric flows

TL;DR

This paper establishes new inequalities linking curvature deviation and isoperimetric ratio for plane curves, and applies these results to analyze the long-term behavior of geometric flows without convexity assumptions.

Contribution

It introduces interpolation inequalities between curvature deviation and isoperimetric ratio, and uses them to study geometric flows of non-convex closed curves.

Findings

Derived inequalities for isoperimetric ratio of plane curves

Analyzed large-time behavior of geometric flows without convexity

Provided tools for studying non-convex curve evolution

Abstract

Several inequalities for the isoperimetric ratio for plane curves are derived. In particular, we obtain interpolation inequalities between the deviation of curvature and the isoperimetric ratio. As applications, we study the large-time behavior of some geometric flows of closed plane curves without a convexity assumption.