Singularities and full convergence of the M\"obius-invariant Willmore flow in the $3$-sphere

Abstract

Here we continue the investigation of the M\"obius-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 M\"obius-transformations of $\mathbb{S}^3$ - as $t \nearrow \infty$.