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

Abstract

In this article the author investigates flow lines of the classical Willmore flow, which start moving in a 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 the threshold $8 \, \sqrt{\frac{\pi^3}{3}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformal image of the standard 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 $\pi:\mathbb{S}^3 \to \mathbb{S}^2$ 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 certain mathematical bridge between the Euler-Lagrange equation of the elastic energy functional and a particular class of elliptic curves over $\mathbb{C}$