ScalingStacks

Proof. [04CJ]

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Proof.

The incline angle of the tangent vector to F⁡(∂Σ)⊂ℂF(\partial\Sigma)\subset\mathbb{C} is equal to the Lagrangian angle modulo π​ℤ\pi\mathbb{Z}. Since L0L_{0} is a special Lagrangian, along the L0L_{0} boundary portion arg⁡F=θ^\arg F=\hat{\theta}. Thus Im​(e−i​θ^​F)=0\text{Im}(e^{-i\hat{\theta}}F)=0 at p,qp,q and the self intersections on L0L_{0}. The Solomon functional integrand simplifies to

Im​∫Σe−i​θ^​F​ω+∑L-self intersections on ∂ΣIm​(e−i​θ^​F)​fL|−+.\text{Im}\int_{\Sigma}e^{-i\hat{\theta}}F\omega+\sum_{\text{$L$-self intersections on $\partial\Sigma$}}\text{Im}(e^{-i\hat{\theta}}F)f_{L}|^{+}_{-}.

By the almost calibrated assumption on L,L0L,L_{0}, and the positivity condition, we obtain Claim 3.23, namely F⁡(Σ)F(\Sigma) lies above its L0L_{0} boundary,

Im​(e−i​θ^​F)≥0on ​Σ.\text{Im}(e^{-i\hat{\theta}}F)\geq 0\quad\text{on }\Sigma.

Morever, the Novikov positivity requirement for the bounding cochain on LL says that fL|−+≥0f_{L}|^{+}_{-}\geq 0 at the degree one self intersections on ∂Σ∩L\partial\Sigma\cap L. Thus the Solomon functional integrand is nonnegative, which implies 𝒮⁡(L)≥0=𝒮⁡(L0)\mathcal{S}(L)\geq 0=\mathcal{S}(L_{0}). ∎

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