Monotonicity of the modulus under curve shortening flow

TL;DR

This paper proves that the modulus of an annulus enclosed by two disjoint curves increases monotonically under curve shortening flow, extending to surfaces with lower curvature bounds.

Contribution

It establishes the monotonicity of the modulus for evolving curves under flow and generalizes the result to surfaces with curvature constraints.

Findings

Modulus of the enclosed annulus increases monotonically over time.

The result applies to curves evolving under curve shortening flow.

Extension of the monotonicity result to surfaces with lower curvature bounds.

Abstract

Given two disjoint nested embedded closed curves in the plane, both evolving under curve shortening flow, we show that the modulus of the enclosed annulus is monotonically increasing in time. An analogous result holds within any ambient surface satisfying a lower curvature bound.