Singularities of the hyperbolic elastic flow: convergence, quantization and blow-ups (Q6607665)

The study of minimizers of Euler's elastic energy is a captivating endeavor at the crossroads of mathematics and physics. It yields a better understanding of the equilibrium states that elastic structures naturally adopt under the influences of external forces, spanning from skyscrapers and bridges to biological tissues. As it turns out, the properties of the elastic flow heavily depend on the geometry of the ambient space. \N\NIn this paper, the authors study the elastic flow of closed curves and of open curves with clamped boundary conditions in the hyperbolic plane. While global existence and convergence toward critical points for initial data with sufficiently small energy is already known, this study pioneers an investigation of the flow's singular behaviour. A convergence theorem is proved, without assuming smallness of the initial energy, and coupled with a quantification of potential singularities. Namely, it is established that each singularity carries an energy cost of at least \(8\). Moreover, the blow-ups of the singularities are explicitly classified. A further contribution is an explicit understanding of the singular limit of the elastic flow of \(\lambda\)-figure-eights, a class of curves that previously served in showing sharpness of the energy threshold 16 for the smooth convergence of the elastic flow of closed curves.