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BY 4.0 license Open Access Published by De Gruyter Open Access March 25, 2020

On the Convergence of the Elastic Flow in the Hyperbolic Plane

  • Marius Müller EMAIL logo and Adrian Spener
From the journal Geometric Flows

Abstract

We examine the L2-gradient flow of Euler’s elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by constructing a class of curves whose lengths blow up in infinite time. The convergence results follow from a constrained sharp Reilly-type inequality.

References

[1] Milton Abramowitz and Irene A. Stegun, Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, vol. 55, 1964. MR 016764210.1115/1.3625776Search in Google Scholar

[2] Christian Bär, Elementary Differential Geometry, Cambridge University Press, 2010.Search in Google Scholar

[3] Matthias Bergner, Anna Dall’Acqua, and Steffen Fröhlich, Symmetric Willmore surfaces of revolution satisfying natural boundary conditions, Calc. Var. Partial Differential Equations 39 (2010), no. 3-4, 361–378. MR 2729304Search in Google Scholar

[4] Simon Blatt, A singular example for the Willmore flow, Analysis (Munich) 29 (2009), no. 4, 407–430. MR 2591055Search in Google Scholar

[5] Simon Blatt, A note on singularities in finite time for the constrained willmore flow, Preprint (2018).Search in Google Scholar

[6] Ralph Chill, Eva Fašangová, and Reiner Schätzle, Willmore blowups are never compact, Duke Math. J. 147 (2009), no. 2, 345–376.Search in Google Scholar

[7] Ralph Chill, On the Łojasiewicz–Simon gradient inequality, Journal of Functional Analysis 201 (2003), no. 2, 572 – 601.Search in Google Scholar

[8] Harold T. Davis, Introduction to nonlinear differential and integral equations, Dover Publications, Inc., New York, 1962. MR 0181773Search in Google Scholar

[9] Manfredo Perdigão do Carmo, Riemannian geometry, Mathematics: Theory & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992. MR 1138207Search in Google Scholar

[10] Anna Dall’Acqua, Klaus Deckelnick, and Hans-Christoph Grunau, Classical solutions to the Dirichlet problem for Willmore surfaces of revolution, Adv. Calc. Var. 1 (2008), no. 4, 379–397. MR 2480063Search in Google Scholar

[11] Gerhard Dziuk, Ernst Kuwert, and Reiner Schätzle, Evolution of elastic curves inn: existence and computation, SIAM J. Math. Anal. 33 (2002), no. 5, 1228–1245. MR 1897710Search in Google Scholar

[12] Anna Dall’Acqua, Tim Laux, Chun-Chi Lin, Paola Pozzi and Adrian Spener, The elastic flow of curves on the sphere, Geometric Flows 3(1), 1–13 (2018).10.1515/geofl-2018-0001Search in Google Scholar

[13] Anna Dall’Acqua and Paola Pozzi, A Willmore-Helfrich L2-flow of curves with natural boundary conditions, Comm. Anal. Geom. 22 (2014), no. 4, 617–669. MR 3263933Search in Google Scholar

[14] Anna Dall’Acqua, Paola Pozzi, and Adrian Spener, The Łojasiewicz–Simon gradient inequality for open elastic curves, Journal of Differential Equations 261 (2016), no. 3, 2168 – 2209.Search in Google Scholar

[15] Anna Dall’Acqua and Adrian Spener, The elastic flow of curves in the hyperbolic plane, Preprint (https://arxiv.org/abs/1710.09600) (2017).Search in Google Scholar

[16] Anna Dall’Acqua and Adrian Spener, Circular solutions to the elastic flow in hyperbolic space, RIMS Kôkyûroku, Proceedings of the conference Analysis on Shapes of Solutions to Partial Differential Equations, Kyoto 2017/06/05–2017/06/07 (2018), no. 2082.Search in Google Scholar

[17] Sascha Eichmann and Hans-Christoph Grunau, Existence for willmore surfaces of revolution satisfying non-symmetric dirichlet boundary conditions, preprint (2016).Search in Google Scholar

[18] Sascha Eichmann, Nichtperiodische Fortsetzbarkeit von Willmore-Flächen unter Axialsymmetrie, 2014.Search in Google Scholar

[19] Sascha Eichmann, Willmore surfaces of revolution satisfying dirichlet data, Preprint (2017).Search in Google Scholar

[20] Ahmad El Soufi and Saïd Ilias, Une inégalité du type “reilly” pour les sous-variétés de l’espace hyperbolique, Commentarii Mathematici Helvetici 67 (1992), no. 1, 167–181.Search in Google Scholar

[21] Alfred Gray, Elsa Abbena, and Simon Salamon, Modern differential geometry of curves and surfaces with Mathematica®, third ed., Studies in Advanced Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 2006. MR 2253203Search in Google Scholar

[22] Mohammad Ghomi, A Riemannian four vertex theorem for surfaces with boundary, Proc. Amer. Math. Soc. 139 (2011), no. 1, 293–303. MR 2729091Search in Google Scholar

[23] Lynn Heller, Constrained Willmore tori and elastic curves in 2-dimensional space forms, Comm. Anal. Geom. 22 (2014), no. 2, 343–369. MR 3210758Search in Google Scholar

[24] Norihito Koiso, Convergence towards an elastica in a Riemannian manifold, Osaka J. Math. 37 (2000), no. 2, 467–487. MR 1427766Search in Google Scholar

[25] Ernst Kuwert and Reiner Schätzle, Removability of point singularities of Willmore surfaces, Ann. of Math. (2) 160 (2004), no. 1, 315–357. MR 2119722Search in Google Scholar

[26] Anders Linnér, Curve-straightening and the Palais-Smale condition, Trans. Amer. Math. Soc. 350 (1998), no. 9, 3743–3765. MR 1432203Search in Google Scholar

[27] Chun-Chi Lin, L2-flow of elastic curves with clamped boundary conditions, J. Differential Equations 252 (2012), no. 12, 6414–6428. MR 2911840Search in Google Scholar

[28] Joel Langer and David Singer, Curves in the hyperbolic plane and mean curvature of tori in 3-space, Bull. London Math. Soc. 16 (1984), no. 5, 531–534. MR 751827Search in Google Scholar

[29] Joel Langer and David Singer, The total squared curvature of closed curves, J. Differential Geom. 20 (1984), no. 1, 1–22. MR 772124Search in Google Scholar

[30] Peter Li and Shing Tung Yau, A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces, Invent. Math. 69 (1982), no. 2, 269–291. MR 674407Search in Google Scholar

[31] Rainer Mandel, Explicit formulas and symmetry breaking for Willmore surfaces of revolution, ArXiv e-prints (2017).10.1007/s10455-018-9598-0Search in Google Scholar

[32] Andrea Mondino and Huy The Nguyen, A gap theorem for Willmore tori and an application to the Willmore flow, Nonlinear Anal. 102 (2014), 220–225. MR 318281010.1016/j.na.2014.02.015Search in Google Scholar

[33] John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 007563910.2307/1969989Search in Google Scholar

[34] Alexander Polden, Curves and surfaces of least total curvature and fourth-order flows, PhD Thesis, Tübingen (1996).Search in Google Scholar

[35] Robert C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52 (1977), no. 4, 525–533. MR 0482597Search in Google Scholar

[36] Daniel Howard Steinberg, Elastic curves in hyperbolic space, ProQuest LLC, Ann Arbor, MI, 1995, Thesis (Ph.D.)–Case Western Reserve University. MR 2693537Search in Google Scholar

[37] Clifford Truesdell, The influence of elasticity on analysis: the classic heritage, Bull. Amer. Math. Soc. (N.S.) 9 (1983), no. 3, 293–310. MR 714991Search in Google Scholar

[38] Thomas J. Willmore, Riemannian Geometry, Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1993. xii+318 pp. ISBN: 0-19-853253-9. MR 1261641Search in Google Scholar

[39] Hassler Whitney, On regular closed curves in the plane, Compositio Math. 4 (1937), 276–284. MR 1556973Search in Google Scholar

[40] Eberhard Zeidler, Nonlinear functional analysis and its applications. II/B, Springer-Verlag, New York, 1990. MR 103349810.1007/978-1-4612-0981-2Search in Google Scholar

Received: 2020-02-12
Accepted: 2020-02-28
Published Online: 2020-03-25

© 2020 Marius Müller et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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