×

zbMATH — the first resource for mathematics

Willmore minimizers with prescribed isoperimetric ratio. (English) Zbl 1288.74027
Summary: Motivated by a simple model for elastic cell membranes, we minimize the Willmore functional among two-dimensional spheres embedded in \({\mathbb R^3}\) with prescribed isoperimetric ratio.

MSC:
74G65 Energy minimization in equilibrium problems in solid mechanics
74K15 Membranes
49Q10 Optimization of shapes other than minimal surfaces
52A40 Inequalities and extremum problems involving convexity in convex geometry
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Berndl K., Lipowsky R., Seifert U.: Shape transformations of vesicles: phase diagrams for spontaneous-curvature and bilayer-coupling models. Phys. Rev. A 44, 1182–1202 (1991) · doi:10.1103/PhysRevA.44.1182
[2] Brakke K.: The Motion of a Surface by its Mean Curvature. Princeton University Press, Princeton (1978) · Zbl 0386.53047
[3] Bryant R.L.: A duality theorem for Willmore surfaces. J. Differ. Geom. 20, 23–53 (1984) · Zbl 0555.53002
[4] Castro-Villarreal, P., Guven, J.: Inverted catenoid as a fluid membrane with two points pulled together. Phys. Rev. E 76 (2007) · Zbl 1110.74038
[5] Deuling H.J., Helfrich W.: Red blood cell shapes as explained on the basis of curvature elasticity. Biophys. J. 16, 861–868 (1976) · doi:10.1016/S0006-3495(76)85736-0
[6] Evans, L.C., Gariepy, R.F.: Measure theory and fine properties of functions. Studies in Advanced Mathematics. CRC Press, Boca Raton, 1992 · Zbl 0804.28001
[7] Gilbarg D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin (2001) · Zbl 1042.35002
[8] Helfrich W.: Elastic properties of lipid bilayers: theory and possible experiments. Zeitschrift für Naturforschung C: J. Biosci. C 28, 693–703 (1973)
[9] Kuwert E., Li Y., Schätzle R.: The large genus limit of the infimum of the Willmore energy. Am. J. Math. 132, 37–51 (2010) · Zbl 1188.53057 · doi:10.1353/ajm.0.0100
[10] Kuwert E., Schätzle R.: Removability of point singularities of Willmore surfaces. Ann. Math. (2) 160, 315–357 (2004) · Zbl 1078.53007 · doi:10.4007/annals.2004.160.315
[11] Li P., Yau S.T.: A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces. Invent. Math. 69, 269–291 (1982) · Zbl 0503.53042 · doi:10.1007/BF01399507
[12] Nagasawa T., Takagi I.: Bifurcating critical points of bending energy under constraints related to the shape of red blood cells. Calc. Var. Partial Differ. Equ. 16, 63–111 (2003) · Zbl 1051.49027 · doi:10.1007/s005260100143
[13] Schygulla, J.: Flächen mit L 2-beschränkter zweiter Fundamentalform nach Leon Simon. Diplomarbeit an der Albert-Ludwigs-Universität Freiburg, 2008
[14] Simon L.: Existence of Surfaces minimizing the Willmore Functional. Commun. Anal. Geom. 1, 281–326 (1993) · Zbl 0848.58012
[15] Simon, L.: Lectures on Geometric Measure Theory. Proceedings of the Centre for Mathematical Analysis, Centre for Mathematical Analysis, Canberra, 1983 · Zbl 0546.49019
[16] Thomsen G.: Über konforme Geometrie I: Grundlagen der konformen Flächentheorie. Hamb. Math. Abh. 3, 31–56 (1923) · JFM 49.0530.02 · doi:10.1007/BF02954615
[17] Willmore, T.: Total curvature in Riemannian Geometry. Ellis Horwood Series: Mathematics and its Applications. Ellis Horwood Ltd., Chichester; Halsted Press [Wiley], New York, 1982 · Zbl 0501.53038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.