Dimension reduction for Willmore flows of tori: fixed conformal class and analysis of singularities

TL;DR

This paper analyzes Willmore flows of tori with fixed conformal class, establishing conditions for singularities, relating symmetric cases to hyperbolic elastic flows, and constructing new conformally constrained tori.

Contribution

It introduces a conformally constrained Willmore flow, relates symmetric cases to hyperbolic elastic flows, and identifies new classes of constrained tori and singularity behaviors.

Findings

Established a necessary condition for singularities.

Linked symmetric Willmore flows to hyperbolic elastic flows.

Constructed new conformally constrained Willmore tori.

Abstract

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 $8\pi$. 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.