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

Minicozzi, William P.

doi:10.1090/S0894-0347-1995-1311825-9

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.

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