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Stationary soap films with vertical potentials. (English) Zbl 1489.53090

Authors’ abstract: We classify cylindrical surfaces in the Euclidean space whose mean curvature is a \(n\)th-power of the distance to a reference plane. The generating curves of these surfaces, called \(n\)-elastic curves, have a variational characterization as critical points of a curvature energy generalizing the classical elastic energy. We give a full description of such curves obtaining, in some particular cases, closed curves including simple ones.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
49Q10 Optimization of shapes other than minimal surfaces
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
37K25 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
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[1] A. D., Alexandrov, Uniqueness theorems for surfaces in the large V, Vestnik Leningrad Univ. Math.. Vestnik Leningrad Univ. Math., AMS Transl., 21, 412-416 (1962), English translation · Zbl 0119.16603
[2] Arroyo, J.; Garay, O. J.; Pámpano, A., Constant mean curvature invariant surfaces and extremals of curvature energies, J. Math. Anal. A, 462, 1644-1668 (2018) · Zbl 1388.49045
[3] Bittencourt, J. E., Fundamentals of Plasma Physics (2004), Springer: Springer New York · Zbl 1084.76001
[4] Blaschke, W., Vorlesungen über Differentialgeometrie und Geometrische Grundlagen von Einsteins Relativitätstheorie I, (Elementare Differentialgeometrie (1921), J. Springer: J. Springer Berlin) · JFM 50.0452.03
[5] Castro, I.; Castro-Infantes, I., Plane curves with curvature depending on distance to a line, Differential Geom. Appl., 44, 77-97 (2016) · Zbl 1353.53007
[6] Delaunay, C., Sur la surface de révolution dont la courbure moyenne est constante, J. Math. Pures Appl., 16, 309-320 (1841)
[7] Euler, L.; Elasticis, De. Curvis., (Methodus Inveniendi Lineas Curvas Maximi Minimive Propietate Gaudentes. Methodus Inveniendi Lineas Curvas Maximi Minimive Propietate Gaudentes, Sive Solutio Problematis Isoperimetrici Lattissimo Sensu Accepti, Additamentum 1 Ser. (1744)), 1 24, Lausanne
[8] Finn, R., Equilibrium capillary surfaces, (Grundlehren Der Math. Wiss. vol. 284 (1986), Springer: Springer New York) · Zbl 0583.35002
[9] Koiso, M.; Palmer, B., Geometry and stability of bubbles with gravity, Indiana Univ. Math. J., 54, 65-98 (2005) · Zbl 1078.53008
[10] Koiso, M.; Palmer, B., On a variational problem for soap films with gravity and partially free boundary, J. Math. Soc. Japan, 57, 333-355 (2005) · Zbl 1072.49029
[11] Langer, J.; Singer, D. A., The total squared curvature of closed curves, J. Differential Geom., 20, 1-22 (1984) · Zbl 0554.53013
[12] Linhart, J. G., Plasma Physics (1960), North-Holland Publishing Co.: North-Holland Publishing Co. Amsterdam · Zbl 0102.23306
[13] López, R., Constant Mean Curvature Surfaces with Boundary (2013), Springer-Verlag Berlin Heidelberg · Zbl 1278.53001
[14] López, R.; Pámpano, A., Classification of rotational surfaces in Euclidean space satisfying a linear relation between their principal curvatures, Math. Nachr., 293, 735-753 (2020) · Zbl 1527.53008
[15] Miura, T., Elastic curves and phase transitions, Math. Ann., 376, 1629-1674 (2020) · Zbl 1436.49060
[16] Mladenov, I.; Hadzhilazova, M., (The Many Faces of Elastica. The Many Faces of Elastica, Forum for Interdisciplinary Mathematics, vol. 3 (2017), Springer) · Zbl 1398.53008
[17] Pámpano, A., Invariant surfaces with generalized elastic profile curves (2018), (Ph.D. thesis)
[18] Serrin, J., A symmetry problem in potential theory, Arch. Ration. Mech. Anal., 43, 304-318 (1971) · Zbl 0222.31007
[19] Singer, D. A., Lectures on elastic curves and rods, (Curvature and Variational Modeling in Physics and Biophysics. Curvature and Variational Modeling in Physics and Biophysics, AIP Conf. Proc., vol. 1002 (2008), American Institute of Physics: American Institute of Physics Melville, NY), 3-32
[20] Truesdell, C., The rational mechanics of flexible or elastic bodies: 1638-1788, (Leonhard Euler, Opera Omnia, Orell Füssli Turici, ser. 2, vol. XI (1960)), 2 · Zbl 0215.31503
[21] Wente, H. C., The symmetry of sessile and pendent drops, Pacific J. Math., 88, 387-397 (1980) · Zbl 0473.76086
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