5.7.1 Special Lagrangians are minimizers [04GL]

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5.7.1 Special Lagrangians are minimizers

We now revisit Prop. 3.40. Our goal is to suggest that the automatic transversality, positivity condition, and even smoothness assumptions can be removed in Prop. 3.40, at the cost of assuming the entire force of the Thomas-Yau conjecture, under the setting of this chapter.

Conjecture 5.25.

If there exists a special Lagrangian $L$ in the class $\mathcal{L}$ , then it is a minimizer of the Solomon functional.

Proof.

(Heuristic) The existence of a special Lagrangian representative should imply Thomas-Yau semistability (cf. Conjecture 3.31). By the Thomas-Yau existence conjecture 5.21 this implies the Solomon functional has a minimizer $L^{\prime}$ , which must be a special Lagrangian. Then the Thomas-Yau uniqueness conjecture 5.24 implies $L=L^{\prime}$ as currents. ∎