For the geometric measure theoretic purpose of finding special Lagrangians, the existence of a minimizer as a Lagrangian current $L$ is probably sufficient. However, for applications to the Fukaya category, it is highly desirable to know that $L$ carries a formal brane structure (cf. section 5.4), which likely requires resolving Question 12. Some analogies suggest the question may be subtle:

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  In geometric invariant theory (GIT), there are niceties concerning semistable, polystable and stable objects. If we take a sequence of semistable objects in a fixed reductive group orbit, the limit may jump outside the orbit, so that the orbit does not admit a polystable representative. Several semistable orbits may be ‘$S$-equivalent’, and each $S$-equivalence class contains a unique polystable orbit.

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  In the gauge theory of holomorphic bundles, likewise a sequence of connections in the same complexified gauge orbit may jump outside the orbit in the limit; algebro-geometrically, this jumping of bundle structure is usually related to bundle extensions.

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  One motivation for the Thomas-Yau program is to form the moduli space of (semi)stable Lagrangian branes. The Hausdorff property of the moduli space is a delicate question.

For these reasons, as well as Remark 5.18, we are not certain if the Lagrangian minimizer should be interpreted as a representative in the chosen $D^{b}Fuk(X)$ class, or if several semistable $D^{b}Fuk(X)$ classes should be identified under some suitable $S$-equivalence relation. We think this question requires further developments in Floer theory. The question is also reflected in the delicacy of the infinite time limit in Joyce’s Bridgeland stability proposal.