Singularities and full convergence of the Möbius-invariant Willmore flow in the $3$-sphere
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2022
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| _version_ | 1866915764788789248 |
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| author | Jakob, Ruben |
| author_facet | Jakob, Ruben |
| contents | 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 $Σ$ 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 $μ$ in $\mathbb{R}^4$, which is the weak limit of the sequence of varifolds $\{\mathcal{H}^2\lfloor_{F_{t_{l}}(Σ)}\}$, for an appropriately chosen sequence $t_{l} \nearrow T_{max}(F_0)$, and that $spt(μ)$ is either empty or homeomorphic to some compact, closed manifold of genus either $0$ or $1$. In the particular case in which $spt(μ)$ 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 $8 π$, 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 $t \nearrow \infty$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_00604 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Singularities and full convergence of the Möbius-invariant Willmore flow in the $3$-sphere Jakob, Ruben Differential Geometry Analysis of PDEs 30C35, 35R01, 53C18, 53C42, 53E40 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 $Σ$ 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 $μ$ in $\mathbb{R}^4$, which is the weak limit of the sequence of varifolds $\{\mathcal{H}^2\lfloor_{F_{t_{l}}(Σ)}\}$, for an appropriately chosen sequence $t_{l} \nearrow T_{max}(F_0)$, and that $spt(μ)$ is either empty or homeomorphic to some compact, closed manifold of genus either $0$ or $1$. In the particular case in which $spt(μ)$ 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 $8 π$, 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 $t \nearrow \infty$. |
| title | Singularities and full convergence of the Möbius-invariant Willmore flow in the $3$-sphere |
| topic | Differential Geometry Analysis of PDEs 30C35, 35R01, 53C18, 53C42, 53E40 |
| url | https://arxiv.org/abs/2205.00604 |