# A distance comparison principle for curve shortening flow with free boundary

## TL;DR

This paper introduces a new geometric comparison principle for free boundary curve shortening flows, enabling the analysis of their long-term behavior and convergence properties within convex planar domains.

## Contribution

It develops a reflected chord-arc profile and establishes a chord-arc estimate for embedded free boundary flows, providing new insights into their asymptotic behavior.

## Key findings

- Flows either converge to a critical chord or contract to a round half-point.
- The method applies to convex planar domains with orthogonal boundary conditions.
- The results extend understanding of free boundary curve evolution.

## Abstract

We introduce a reflected chord-arc profile for curves with orthogonal boundary condition and obtain a chord-arc estimate for embedded free boundary curve shortening flows in a convex planar domain. As a consequence, we are able to prove that any such flow either converges in infinite time to a (unique) ``critical chord'', or contracts in finite time to a ``round half-point'' on the boundary.