Non-orientable Lagrangian surfaces with controlled area (Q1598403)

The author considers the \(4\)-dimensional standard symplectic vector space \((R^4,\omega=d\eta)\). Then the symplectic area of any piecewise smooth closed curve \(\gamma\) in \(R^4\) is defined to be \(\int_{\gamma}\eta\). Considering two curves \(\gamma_1\) and \(\gamma_2\) such that the sum of their symplectic areas is zero, the author proves the existence of an oriented Lagrangian surface having \(\gamma_1\) and \(\gamma_2\) as boundary and such that the area of that surface can be controlled by expressions in \(\gamma_1\) and \(\gamma_2\). As an immediate consequence, any closed curve bounds a non-oriented Lagrangian surface such that the area is bounded by the length square of the given curve. Finally, this result is extended to flat chains mod 2. This implies that the set of all non-oriented Lagrangian surfaces is dense in the space of all non-oriented surfaces in \(R^4\).