The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers
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- by William P. Minicozzi
- J. Amer. Math. Soc. 8 (1995), 761-791
- DOI: https://doi.org/10.1090/S0894-0347-1995-1311825-9
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Abstract:
In this paper we prove an existence and regularity theorem for lagrangian tori minimizing the Willmore functional in Euclidean four-space, ${{\mathbf {R}}^4}$, with the standard metric and symplectic structure. Technical difficulties arise because the Euler-Lagrange equation for this problem is a sixth-order nonlinear partial differential equation. This research was motivated by a study of the seemingly unrelated Plateau problem for lagrangian tori, and in this paper we illustrate this connection.References
- Bang-yen Chen, Geometry of submanifolds, Pure and Applied Mathematics, No. 22, Marcel Dekker, Inc., New York, 1973. MR 0353212
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0 M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
- David A. Hoffman, Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geometry 8 (1973), 161–176. MR 390973
- 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 674407, DOI 10.1007/BF01399507
- Charles B. Morrey Jr., Multiple integrals in the calculus of variations, Die Grundlehren der mathematischen Wissenschaften, Band 130, Springer-Verlag New York, Inc., New York, 1966. MR 0202511, DOI 10.1007/978-3-540-69952-1 R. Schoen and J. Wolfson (in preparation).
- Leon Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), no. 2, 281–326. MR 1243525, DOI 10.4310/CAG.1993.v1.n2.a4
- Leon Simon, Lectures on geometric measure theory, Proceedings of the Centre for Mathematical Analysis, Australian National University, vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra, 1983. MR 756417
- James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 233295, DOI 10.2307/1970556
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- François Trèves, Basic linear partial differential equations, Pure and Applied Mathematics, Vol. 62, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1975. MR 0447753
- Joel L. Weiner, On a problem of Chen, Willmore, et al, Indiana Univ. Math. J. 27 (1978), no. 1, 19–35. MR 467610, DOI 10.1512/iumj.1978.27.27003
- Thomas J. Willmore, Total curvature in Riemannian geometry, Ellis Horwood Series: Mathematics and its Applications, Ellis Horwood Ltd., Chichester; Halsted Press [John Wiley & Sons, Inc.], New York, 1982. MR 686105
Bibliographic Information
- © Copyright 1995 American Mathematical Society
- Journal: J. Amer. Math. Soc. 8 (1995), 761-791
- MSC: Primary 58E12; Secondary 35J60, 49Q05, 53C42
- DOI: https://doi.org/10.1090/S0894-0347-1995-1311825-9
- MathSciNet review: 1311825