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

Anna Dall'Acqua, Marius Müller, Fabian Rupp, Manuel Schlierf

TL;DR

This work studies Willmore flows of tori under a fixed conformal class, establishing a one-dimensional reduction to hyperbolic elastic energy for rotationally symmetric data. The authors develop a constrained gradient framework, derive weighted curvature estimates, and prove convergence to conformally constrained Willmore tori, even for initial data with energy beyond the usual $8\pi$ threshold, by leveraging a constrained Łojasiewicz–Simon gradient inequality. A key contribution is identifying the inverted catenoid as a non-smooth singular limit for the Willmore flow of certain tori and showing how to restart the flow from this surface to converge to a round sphere. The results yield new classes of conformally constrained Willmore tori, provide a detailed analysis of singularity formation, and offer a rigorous pathway toward surgery-like handling of Willmore flows and potential weak solutions in higher genus settings.

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 $8π$. 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.

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

TL;DR

This work studies Willmore flows of tori under a fixed conformal class, establishing a one-dimensional reduction to hyperbolic elastic energy for rotationally symmetric data. The authors develop a constrained gradient framework, derive weighted curvature estimates, and prove convergence to conformally constrained Willmore tori, even for initial data with energy beyond the usual threshold, by leveraging a constrained Łojasiewicz–Simon gradient inequality. A key contribution is identifying the inverted catenoid as a non-smooth singular limit for the Willmore flow of certain tori and showing how to restart the flow from this surface to converge to a round sphere. The results yield new classes of conformally constrained Willmore tori, provide a detailed analysis of singularity formation, and offer a rigorous pathway toward surgery-like handling of Willmore flows and potential weak solutions in higher genus settings.

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 . 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.

Paper Structure

This paper contains 24 sections, 48 theorems, 249 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Conformally constrained minimization and Willmore flow
  3. The Willmore flow with fixed conformal class
  4. Analysis of singularities
  5. Structure of this article
  6. Geometric preliminaries
  7. Tori of revolution and hyperbolic geometry
  8. Evolution equations and the Willmore flow
  9. The conformally constrained Willmore gradient and flow
  10. Weighted curvature estimates
  11. Interpolation estimates of Gagliardo--Nirenberg type
  12. Energy estimates
  13. Controlling the non-local Lagrange multiplier
  14. Proof of convergence
  15. Life-span bound via the distance to the rotation axis
  16. ...and 9 more sections

Key Result

Theorem 1

Let $f_0\colon\mathbb{T}^2\to\mathbb{R}^3$ be a rotationally symmetric torus. Then there exists a maximal solution $f\colon[0,T)\times\mathbb{T}^2\to\mathbb{R}^3$ to the conformally constrained Willmore flow such that at least one of the following is true. Moreover, if $\mathcal{W}(f_0)\leq 8\pi$ or if the initial profile curve has hyperbolic length $L$ and turning number $m\in\mathbb{Z}$ such th

Figures (2)

  • Figure 1: Plots of $\lambda$-figure eights for $\lambda\in\{0.5,0.3,0.005\}$ and associated elastic energy $\mathcal{E}_{\lambda}$ close to $16$, all having a self-intersection at $(0,1)$.
  • Figure 2: Illustrations for \ref{['lem:wf-starting-in-ic-1']} and its proof.

Theorems & Definitions (103)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • Lemma 2
  • Definition 1
  • Proposition 2.1
  • proof
  • Definition 2
  • ...and 93 more