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Global Existence and Full Convergence of the Möbius-Invariant Willmore Flow in the 3-Sphere

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Abstract

In this article, we prove two global existence and full convergence theorems for flow lines of the Möbius-invariant Willmore flow, and we use the latter result in order to prove that fully and smoothly convergent flow lines of the Möbius-invariant Willmore flow are stable with respect to small perturbations of their initial immersions in any \(C^{4,\gamma }\)-norm, provided they converge to a umbilic-free \(C^4\)-local minimizer of the Willmore functional among \(C^4\)-immersions of a smooth compact torus into either \({\textbf{R}}^3\) or \({\textbf{S}}^3\). The proofs of our two main theorems rely on the author’s recent achievements about the Möbius-invariant Willmore flow, on Weiner’s investigation of the stability of the Clifford torus with respect to the Willmore functional, and on Escher’s, Mayer’s, and Simonett’s work from the 1990 s on invariant center manifolds for uniformly parabolic quasilinear evolution equations and their special applications to the Willmore- and surface diffusion flow near round 2-spheres in \({\textbf{R}}^3\).

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Notes

  1. See Definition 2 (b), (d) and (e) in [19], where the terms “flow line”, “life span” and “singular time” have been introduced.

  2. See here Sect. 5.1 in [36] for a very nice exposition.

  3. See here Theorem 5.1 in [25].

  4. Compare here with our Lemma 3 and also with the proof of our Lemma 4 below.

  5. Here and in the sequel, “full convergence of a flow line” \(\{f_t\}_{t \in [0,\infty )}\) of either evolution equation (2) or equation (3) in a Banach space \((X,\parallel \cdot \parallel _{X})\) means that \(\parallel f_t - f^* \parallel _{X} \longrightarrow 0\) for a unique limit \(f^* \in X\), as \(t \rightarrow \infty \), in contrast to “subconvergence” of \(\{f_t\}_{t \in [0,\infty )}\) with respect to \(\parallel \cdot \parallel _{X}\), which means that only for certain sequences \(t_j \nearrow \infty \) the immersions \(f_{t_j}\) converge to certain limits in X, that might in general depend on those special sequences \(\{t_j\}\).

  6. See here also Sect. 4.1 in [35] for the construction of the map X in Euclidean space, and especially formulae (27) and (34) in [35] for a simple upper bound on a in terms of the maximal principle curvature of the base surface, which is here the Clifford torus \({\mathcal {C}} \hookrightarrow {\textbf{S}}^3\).

  7. We can easily infer from formula (5) in [48] that—on the one hand—the values on both sides of (22) and (23) do not change if we flip the unit normal from \(\nu _{\varXi (\rho _t)}\) to \(-\nu _{\varXi (\rho _t)}\). On the other hand, in equation (25) we have simply dropped the unit normal \(\nu _{\varXi (\rho _t)}\) appearing in equations (22) and (23). Therefore the ±-ambiguity in (16), leading to \(V(x,t)=\pm |(\partial _t)^{\perp _{X}}(X(x,\rho (x,t)))|\) on account of (17), disappears in (25), if we choose a continuous field of unit normals \(\nu _{\varXi (\rho _t)}\) along the moving surfaces \(\varXi (\rho _t)\) in such a way that \(\nu _{\varXi (\rho _t)}= \nu _{{\textbf{C}}}\) holds for \(\rho _t \equiv 0\), just as asserted in [38], formula (1.1), or in [39], formula (5.1).

  8. Here our procedure clearly differs from the one in [13] and [44], where the authors relied on the computations in Lemma 3.1 of [14], which we only used in the proof of Lemma 1.

  9. Compare here also with p.31 in [48], with Appendix B in [33] or with Sect. 6 in [46].

  10. Compare here also with Proposition B.1 in [33].

  11. Compare here with the statements of Lemma 3.2 in [48].

  12. This result is not new, and it was stated without proof on page 34 in [48].

  13. See here our construction of Fermi coordinates about \({\mathcal {C}}\) in (8)–(10) and formula (35) in [35] for the similar case in which the ambient space is \({\textbf{R}}^3\).

  14. See here also the second part of Theorem 5 below.

  15. See here also Sect. 2 in [12], Sect. 3 in [28] and [11].

  16. See here Definition 4.1.1 in [27].

  17. See here also Definition 2.0.1 in [27].

  18. See here pp. 88–89 in [4] for an exact definition of parabolic \(L^p\)-function spaces.

  19. It should be stressed here that this theorem indeed holds already for \(C^{\infty }\)-smooth, umbilic-free initial immersions.

  20. Compare here also with p. 1156 in [17] and with formula (12) in [19], where the tensorial character of the expressions \((\varGamma _{F_0})^m_{kl} - (\varGamma _{f})^m_{kl}\) and \({\mathcal {T}}_{F_0}(f)\) had been explained.

  21. Compare here with Definition 2 (d) in [19], introducing the life span of a flow line.

  22. The interested reader might also want to compare the first two parts of our Proposition 1 with the first three parts of Theorem 1 in [19], whose proofs are based on formulae (24)–(28) within the preparatory Lemma 1 in [19], where the quasilinear structure of the non-linear operator \([f \mapsto {\mathcal {M}}_{F_0}(f)]\) from lines (127), (128) and (131) above has been precisely analyzed.

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Acknowledgements

The author was funded by the Ministry of Absorption of the State of Israel in the academic years 2019/2020–2021/2022. Moreover, he would like to thank firstly Professor Itai Shafrir and Professor Yehuda Pinchover for their hospitality and strong scientific support at the Mathematics Department of the “Israel Institute of Technology” and secondly the diligent referees for their invaluable corrections, remarks and questions.

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    Ruben Jakob

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Jakob, R. Global Existence and Full Convergence of the Möbius-Invariant Willmore Flow in the 3-Sphere. J Geom Anal 34, 24 (2024). https://doi.org/10.1007/s12220-023-01464-x

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