Compactness of the unpenalized area-preserving elastic flow for closed curves

Establish compactness for the area-preserving elastic flow of closed curves in the absence of a length penalization.

Background

The paper contrasts the area-preserving curvature flow studied here with the area-preserving elastic flow, which is the H−1H^{-1}-gradient flow of the bending energy. For closed curves, prior work established global-in-time existence and compactness when an additional length penalization is imposed. The unresolved case is compactness without that penalization, a difficulty that motivates studying open curves with endpoints constrained to skew lines, where suitable boundary conditions may provide additional geometric control.

References

In the absence of such a penalization, however, compactness for the area-preserving elastic flow of closed curves remains an open problem.

— Area-preserving curvature flow for open curves with general contact angles on skew lines  (2610.05909 - Hiroi, 5 Oct 2026) in Section 1, Introduction