Let $M$ be (for example) a Calabi-Yau threefold with Kaehler form $\omega$ and holomorphic 3-form $\Omega$. We say that a submanifold $L$ of $M$ is a special Lagrangian submanifold if $L$ is Lagrangian with respect to symplectic form $\omega$ and also $\mathrm{Im}\Omega|_L=0$. I would like to know geometric intuition of the latter condition. I am aware that Lagrangian condition roughly corresponds to $L$ having only position coordinate, momentum coordinate, or certain mixture of them (Lagrangian formulation of classical mechanics). I wonder if there is a good explanation about the extra condition $\mathrm{Im}\Omega|_L=0$.
[Math] geometric intuition of special Lagrangian manifolds
ag.algebraic-geometrysg.symplectic-geometry
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In some sense this really goes back to pre-index theory days to Vekua and was one of the motivations for the index theorem (for a reference to Vekua see Gromov's psuedo-holomorphic curves paper and I think there is long discussion in Booss and Bleecker). Vekua proved the following. Take a map from $f:S^1 \to \mathbb{C}^*$. Consider the operator $u \mapsto \bar \partial u + a u + b \bar u$ on the domain $ u\in C^\infty(D,\mathbb{C})$ so that $ Re(\bar f u|{\partial D})=0 $ mapping to $C^\infty(D,\mathbb{C})$. This operator has index equal to the degree of $f$. You can translate this to a Maslov index. It forces the boundary values of $f$ to lie in this line which spins around according to the degree of $f$. The index is really rather easy to compute once you prove that this is a Fredholm boundary value problem. To compute the index, use homotopy invariance of the index to reduce to a model computation so that $f$ is homotopic to $z^k$ for some $k \in \mathbb{Z}$ and $a=b=0$. Then the kernel consist of holomorphic functions $u$ on the disk which satisfy the boundary condition, if $k\ge 0$ these are the real span of the polynomials $iz^k, z^{k-1}-z^{k+1},z^{k-2}-z^{k+2},\ldots$ so that its dimension is $k+1$. For $k \ge 0$ the kernel is zero. The cokernel consists of anti-holomorphic $v$ functions which satisfy the adjoint boundary condition $Im(f v|_{\partial D^2})=0$. For $k\le 0$ a real basis is given by $\bar z^k, \bar z^{k-1}+\bar z^{k+1},\bar z^{k-2}+\bar z^{k+2},\ldots$. Thus the index is $k$ no mater the sign of $k$.
Symplectic geometry: The natural geometry of economics? (Thomas Russel, 2011).
What restrictions does the hypothesis of maximizing behavior place on observed market data? In the context of profit maximization, one of the conditions put forward by Samuelson is a ratio test for the areas between two restricted input demand functions. Here we place this condition firmly within the context of modern mathematics and will indicate why area geometry, (or in higher dimensions, symplectic geometry) seems to be the natural geometry of maximizing economics.
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Best Answer
Let $(M,g,J,\Omega)$ be a Calabi-Yau $n$-fold. Then $\textrm{Re }\Omega$ is a calibration on $(M,g)$. Let $L\subset M$ be a real submanifold with $\dim_{\mathbb{R}}L=n$. You have the following
Proposition
$L$ is a special Lagrangian if and only if it admits an orientation making it into a calibrated (for $\textrm{Re }\Omega$) submanifold of $(M,g)$. In that case, it is volume-minimising in its homology class.
See Propositions 10.1 and 7.1 in the "Calabi-Yau manifolds..." book by Gross, Huybrechts and Joyce.
Here $\textrm{Re }\Omega$ being a calibration means that at each point $p\in M$, and for every oriented tangent $n$-plane $V\subset T_{M,p}$, one has $\left.\textrm{Re }\Omega\right|_V\leq vol_V$, where $vol$ is the volume form of $g$. Then $L$ being a calibrated submanifold for $\textrm{Re }\Omega$ means that on the tangent spaces $T_{L,p}$ of $L$ the above inequality becomes equality.