A Curvature Flow and Applications to an Isoperimetric Inequality

TL;DR

This paper proves long-term existence and convergence of a curvature flow for radial graphs in warped surfaces, using it to establish a general isoperimetric inequality under weak conditions.

Contribution

It introduces a curvature flow approach for radial graphs in warped product surfaces and applies it to derive a new isoperimetric inequality.

Findings

Flow exists long-term and converges to a circle

Flow preserves enclosed area

Derived a general isoperimetric inequality

Abstract

Long time existence and convergence to a circle is proved for radial graph solutions to a mean curvature type curve flow in warped product surfaces (under a weak assumption on the warp potential of the surface). This curvature flow preserves the area enclosed by the evolving curve, and this fact is used to prove a general isoperimetric inequality applicable to radial graphs in warped product surfaces under weak assumptions on the warp potential.