2025/02/18 by Anna Dall’Acqua, Marius Müller, Dall'Acqua, Anna +5 · 1 citation
Engineering · #35B40 #49Q10 (secondary) #53E40 (primary) #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #Fluid Dynamics and Turbulent Flows
paper · pdf · doi:10.48550/arxiv.2502.12606
openalex publication_date 2025/02/18 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
This work studies Willmore flows of tori and their singularities via a dimension reduction approach. We introduce a Willmore flow that preserves the degenerate constraint of prescribed conformal class and, for rotationally symmetric initial data, we establish a strong relation with the length-preserving elastic flow in the hyperbolic plane. We provide a necessary condition for singularities and a criterion for the initial datum that allows to exclude them. Our results allow for initial data with arbitrarily large energy, in particular exceeding the usual Li-Yau threshold of 8π. As an application, we obtain existence of a new class of conformally constrained Willmore tori. Moreover, we investigate singularities of the classical Willmore flow. For a class of tori, we identify a non-smooth object, the inverted catenoid, as the limit shape and we show that the flow can be restarted at this singular surface and converges to a round sphere.