Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains

We classify the convex ancient solutions to the free boundary curve shortening flow in compact, convex domains. We do this using relatively soft methods and the classification comes in two parts:

1. Existence: This is done by explicitly constructing an ancient solution as a limit of so called old-but-not-ancient solutions.
2. Uniqueness: Once it is known that a convex ancient solution in this setting must converge backwards in time to a static chord, uniqueness may be established by demonstrating the indifference of the quantity \(\displaystyle\lim_{t \to -\infty}e^{-\lambda_{0}^{2}t}y(x,t)\) on the ancient solution.

The journal article may be found here.

Recommended citation: Theodora Bourni, Nathan Burns and Spencer Catron (2025). "Classification of Convex Ancient Solutions to Free Boundary Curve Shortening Flow in Convex Domains." The Journal of Geometric Analysis. 35(210).
Download Paper | Download Bibtex