The Willmore flow of Hopf-tori in the $3$-sphere
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arXiv
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| Natura: | Preprint |
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2020
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| author | Jakob, Ruben |
| author_facet | Jakob, Ruben |
| contents | In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth 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 or equal to the threshold $\frac{8π^2}{\sqrt{2}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformally transformed 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 $π:\mathbb{S}^3 \longrightarrow \mathbb{S}^2$ 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 well-known classification and description of smooth, arc-length parametrized solutions of the Euler-Lagrange equation of the elastic energy functional in terms of Jacobi Elliptic Functions and Elliptic Integrals, dating back to the 80s. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2002_01006 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | The Willmore flow of Hopf-tori in the $3$-sphere Jakob, Ruben Analysis of PDEs 53C42, 53E40, 35R01, 58J35, 11Z05 G.0 In this article, the author investigates flow lines of the classical Willmore flow, which start to move in a smooth 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 or equal to the threshold $\frac{8π^2}{\sqrt{2}}$, then the unique flow line of the Willmore flow, starting to move in $F_0$, converges fully to a conformally transformed 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 $π:\mathbb{S}^3 \longrightarrow \mathbb{S}^2$ 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 well-known classification and description of smooth, arc-length parametrized solutions of the Euler-Lagrange equation of the elastic energy functional in terms of Jacobi Elliptic Functions and Elliptic Integrals, dating back to the 80s. |
| title | The Willmore flow of Hopf-tori in the $3$-sphere |
| topic | Analysis of PDEs 53C42, 53E40, 35R01, 58J35, 11Z05 G.0 |
| url | https://arxiv.org/abs/2002.01006 |