| Popis: |
In this article we continue our investigation of the M\"obius-invariant Willmore flow, starting in arbitrary smooth and umbilic-free initial immersions $F_0$ which map some fixed compact torus into $\mathbb{R}^n$ respectively $\mathbb{S}^n$. We investigate the behavior of flow lines $\{F_t\}$ of the M\"obius-invariant Willmore flow in $\mathbb{S}^3$ starting with Willmore energy below $8\pi$, as the time approaches the maximal time of existence $T_{max}(F_0)$ of $\{F_t\}$. We particularly investigate the formation of singularities of flow lines of the M\"obius-invariant Willmore flow in both finite and infinite time. We will see that every limit surface of a flow line of the MIWF can be identified with the support of a certain integral $2$-varifold $\mu$ in $\mathbb{R}^4$ and that the support of this varifold $\mu$ has either degenerated to a point or is homeomorphic to either a sphere or a compact torus in $\mathbb{S}^3$. In the non-degenerate case in which the limit surface is a compact torus, it can be parametrized by a uniformly conformal bi-Lipschitz homeomorphism of class $(W^{2,2}\cap W^{1,\infty})(\Sigma)$. Under certain additional conditions on the considered flow line, such a non-degenerate limit surface of $\{F_t\}$ can be parametrized by a uniformly conformal diffeomorphism of class $W^{4,2}(\Sigma,\mathbb{R}^4)$. Finally, if the initial immersion $F_0$ of a flow line is assumed to parametrize a smooth Hopf-torus with Willmore energy smaller than $8\pi$, then we obtain stronger types of convergence of particular subsequences of $\{F_t\}$ to uniformly conformal $W^{4,2}$-parametrizations of certain limit Hopf-tori, and this insight yields a simple criterion for full $C^m$-convergence of such a flow line of the MIWF to a smooth parametrization of the Clifford-torus - up to a M\"obius-transformation of $\mathbb{S}^3$ - for every fixed $m \in \mathbb{N}$. |
