The Willmore flow of Hopf-tori in the $3$-sphere

TL;DR

This paper proves that the Willmore flow of Hopf-tori in the 3-sphere exists globally without singularities and converges to a Clifford torus under certain energy conditions, using symmetry and elliptic function techniques.

Contribution

It establishes global existence and convergence of the Willmore flow for Hopf-tori, extending understanding of geometric flows in symmetric settings with energy thresholds.

Findings

Flow lines exist globally and do not develop singularities.

Flow lines subconverge to smooth Willmore-Hopf-tori.

Under energy threshold, flow converges to a Clifford torus.

Abstract

In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth parametrization of a Hopf-torus in $\mathbb{S}^3$. We prove that any such flow line of the Willmore flow exists globally, in particular does not develop any singularities, and subconverges to some smooth Willmore-Hopf-torus in every $C^{m}$-norm. Moreover, if in addition the Willmore energy of the initial immersion $F_0$ is required to be smaller than or equal to the threshold $\frac{8\pi^2}{\sqrt{2}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformally transformed Clifford torus in every $C^{m}$-norm, up to time dependent, smooth reparametrizations. Key instruments for the proofs are the equivariance of the Hopf-fibration $\pi:\mathbb{S}^3 \longrightarrow \mathbb{S}^2$ w.r.t. the effect of the $L^2$-gradient of the Willmore energy applied to smooth Hopf-tori in $\mathbb{S}^3$ and to smooth closed regular curves in $\mathbb{S}^2$, a particular version of the Lojasiewicz-Simon gradient inequality, and a well-known classification and description of smooth, arc-length parametrized solutions of the Euler-Lagrange equation of the elastic energy functional in terms of Jacobi Elliptic Functions and Elliptic Integrals, dating back to the 80s.