Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 

 

The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers


Author: William P. Minicozzi
Journal: J. Amer. Math. Soc. 8 (1995), 761-791
MSC: Primary 58E12; Secondary 35J60, 49Q05, 53C42
MathSciNet review: 1311825
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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 [Enhancements On Off] (What's this?)

  • [1] Bang-yen Chen, Geometry of submanifolds, Marcel Dekker, Inc., New York, 1973. Pure and Applied Mathematics, No. 22. MR 0353212
  • [2] 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
  • [3] M. Gromov, Pseudo-holomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), 307-347.
  • [4] David A. Hoffman, Surfaces of constant mean curvature in manifolds of constant curvature, J. Differential Geometry 8 (1973), 161–176. MR 0390973
  • [5] 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, 10.1007/BF01399507
  • [6] 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
  • [7] R. Schoen and J. Wolfson (in preparation).
  • [8] Leon Simon, Existence of surfaces minimizing the Willmore functional, Comm. Anal. Geom. 1 (1993), no. 2, 281–326. MR 1243525, 10.4310/CAG.1993.v1.n2.a4
  • [9] 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
  • [10] James Simons, Minimal varieties in riemannian manifolds, Ann. of Math. (2) 88 (1968), 62–105. MR 0233295
  • [11] Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • [12] François Trèves, Basic linear partial differential equations, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 62. MR 0447753
  • [13] Joel L. Weiner, On a problem of Chen, Willmore, et al, Indiana Univ. Math. J. 27 (1978), no. 1, 19–35. MR 0467610
  • [14] 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

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 58E12, 35J60, 49Q05, 53C42

Retrieve articles in all journals with MSC: 58E12, 35J60, 49Q05, 53C42


Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1995-1311825-9
Article copyright: © Copyright 1995 American Mathematical Society