The elastic flow of curves in the hyperbolic plane

Abstract

We consider closed curves in the hyperbolic space moving by the $L^2$-gradient flow of the elastic energy and prove well-posedness and long time existence. Under the additional penalisation of the length we show subconvergence to critical points. A motivation for the study of this flow is given by the relation between elastic curves in the hyperbolic plane and Willmore surfaces of revolution.

40 pages, submitted to Communications in Analysis and Geometry