Geodesic
The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers
Journal of the American Mathematical Society, Tome 08 (1995) no. 4, pp. 761-791

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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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Minicozzi, William P. The Willmore functional on Lagrangian tori: its relation to area and existence of smooth minimizers. Journal of the American Mathematical Society, Tome 08 (1995) no. 4, pp. 761-791. doi: 10.1090/S0894-0347-1995-1311825-9

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