= Lagrangian path-area functional
{c}
{title2=$\mathcal A(\gamma)=-\int u^*\omega$}
Choose a reference path between <Lagrangian submanifolds> $L_0,L_1$ in a <symplectic manifold>. With a connecting homotopy strip oriented by $ds\wedge dt$, define $\mathcal A(\gamma)=-\int u^*\omega$. If the <path space> is connected and $H_2(M,L_0\cup L_1)=0$, <relative symplectic-area independence> makes this single-valued up to a global constant. Its derivative is $d\mathcal A(\xi)=\int\omega(\dot\gamma,\xi)dt$, so its critical points are the constant paths in $L_0\cap L_1$. For a <compatible almost complex structure>, its formal downward $L^2$ flow is the <J-holomorphic curve> equation $u_s+Ju_t=0$.
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