On the isoperimetric inequality and surface diffusion flow for multiply winding curves

TL;DR

This paper proves a generalized isoperimetric inequality for symmetric closed curves in the plane and applies it to establish global existence of surface diffusion flow for curves close to multiply covered circles.

Contribution

It introduces a new isoperimetric inequality for symmetric curves and demonstrates its use in proving global existence of surface diffusion flow for specific initial conditions.

Findings

Generalized isoperimetric inequality for symmetric curves

Global existence of surface diffusion flow near multiply covered circles

Applicability to non-convex immersed curves

Abstract

In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application we obtain a global existence result for the surface diffusion flow, providing that an initial curve is $H^2$-close to a multiply covered circle and sufficiently rotationally symmetric.