Singularities and full convergence of the Möbius-invariant Willmore flow in the $3$-sphere

Abstract

Here we continue the investigation of the Möbius-invariant Willmore flow (MIWF), starting to move in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus $\Sigma$ into ${\mathbb{R}}^n$ respectively ${\mathbb{S}}^n$. Here we investigate the behaviour of flow lines $\{F_t\}$ of the MIWF in ${\mathbb{S}}^3$ starting with relatively low Willmore energy, as the time $t$ approaches the maximal time of existence $T_{max}(F_0)$ of $\{F_t\}$. We succeed to construct divergent flow lines, and we investigate limit surfaces of both divergent and convergent flow lines of the MIWF. At least generically a limit surface of some general flow line $\{F_t\}$ of the MIWF can be identified with the support of an integral $2$-varifold $\mu$ in ${\mathbb{R}}^4$, which is the weak limit of the sequence of varifolds $\{{\mathcal{H}}^2{\lfloor }_{F_{t_l}(\Sigma )}\}$, for an appropriately chosen sequence $t_l\nearrow T_{max}(F_0)$, and that $spt(\mu )$ is either empty or homeomorphic to some compact, closed manifold of genus either $0$ or $1$. In the particular case in which $spt(\mu )$ is a compact surface of genus $1$ it can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism $f$ of class $W^{2,2}\cap W^{1,\infty }$, and under certain additional conditions on $\{F_{t_l}\}$ such a parametrization is a diffeomorphism of class $W^{4,2}$. Finally, if the initial immersion $F_0$ of a flow line $\{F_t\}$ is assumed to parametrize a Hopf-torus in ${\mathbb{S}}^3$ with Willmore energy not bigger than $8\pi$, then we obtain more precise statements about the flow line $\{F_t\}$ as $t\nearrow T_{max}(F_0)$. This insight will finally yield a criterion for full convergence of such flow lines of the MIWF to parametrizations of the Clifford torus - up to Möbius-transformations of ${\mathbb{S}}^3$ - as $t\nearrow \infty$.