3.8.2 Lower bound of the Solomon functional [04CH]

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3.8.2 Lower bound of the Solomon functional

The theme of Chapter 5 will be on the variational approach to find special Lagrangians by minimizing the Solomon functional in a fixed derived category class. As an important motivation, special Lagrangians are formal local minimizers of the Solomon functional under Hamiltonian deformations (cf. section 2.8). In fact we can do better under the automatic transversality and the positivity condition:

Proposition 3.40.

(‘special Lagrangians are minimizers’) Suppose $L_{0}$ is an exact immersed special Lagrangian of phase $\hat{\theta}\in(-\frac{\pi}{2},\frac{\pi}{2})$ , with unobstructed bounding cochain structure. Let $L$ be an almost calibrated, exact, immersed Lagrangian in the same $D^{b}Fuk(X)$ class, which intersects $L_{0}$ transversely. Suppose the bordism current $\mathcal{C}$ with $\partial\mathcal{C}=L-L_{0}$ satisfies automatic transversality and the positivity condition. Then $\mathcal{S}(L)\geq\mathcal{S}(L_{0})$ .

Proof.

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

\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 $L,L_{0}$ , and the positivity condition, we obtain Claim 3.23, namely $F(\Sigma)$ lies above its $L_{0}$ boundary,

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

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

Remark 3.20.

Suppose we drop the positivity condition, then the key step $\text{Im}(e^{-i\hat{\theta}}F)\leq 0$ would break down, so the above proof of Prop. 3.40 would be invalidated. However, the conclusion may still be true (cf. section 5.7.1).