Non-orientable Lagrangian Surfaces with Controlled Area

We show that any closed curve in $\mathbb{R}^{4}$ bounds a Lagrangian Möbius band with quadratic area(i.e. area bounded by length square). And we generalize this result to flat chains $\bmod 2$ to conclude that in $\mathbb{R}^{4}$ any one-dimensional integral flat chain $\bmod 2$ without boundary bounds a two-dimensional Lagrangian integral flat chain $\bmod 2$ with quadratic area. Moreover we prove that in $\mathbb{R}^{4}$ the set of Lagrangian integral flat chains $\bmod 2$ is dense under the flat norm in the space of all two-dimensional integral flat chains $\bmod 2$.

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Published 1 January 2001