The free-boundary curve shortening flow evolves a curve inside a convex domain, with endpoints free to slide along the boundary, shrinking the curve by its curvature at every point. Under this flow, a semi-circular arc in a convex domain contracts to a round half-point in finite time. The sharp convergence rate — meaning the estimate cannot be improved — characterizes exactly how fast the curve approaches its terminal shape as the singular time approaches.
Sharp rates in geometric flows are rare and structurally informative. A non-sharp estimate says the flow converges; a sharp estimate says there is no hidden slack in the convergence — the system approaches its limit as fast as geometry permits. The sharpness of the rate here means that the semi-circle is, in a precise sense, the most efficient path to extinction for this class of free-boundary evolutions. Any perturbation either converges at the same rate or slower.
The general principle: in systems that evolve toward a singularity, the shape that achieves the singularity most efficiently reveals the geometry's intrinsic preference. The semi-circle is not just one possible extinction profile — it is the attractor, the shape that the flow's internal logic selects. Optimality under extinction exposes the flow's hidden symmetry.
(arXiv:2603.06949)