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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
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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.

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Article copyright: © Copyright 1995 American Mathematical Society