5.5.1 Thomas-Yau conjecture [04G6]

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5.5.1 Thomas-Yau conjecture

The following is our interpretation of the Thomas-Yau existence conjecture:

Conjecture 5.21.

Let $L_{0}$ be an exact, quantiatively almost calibrated, unobstructed Lagrangian object in $\mathcal{L}$ . Assuming Thomas-Yau semistability for $L_{0}$ , then the following (equivalent) statements hold:

1.

There is a special Lagrangian representative in $\mathcal{L}$ .
2.

There is no distinguished triangle in $\mathcal{L}$ satisfying the destabilizing condition.
3.

The Solomon functional is bounded from below on $\mathcal{L}$ .
4.

The Solomon functional has a minimizer in $\mathcal{L}$ .

Here is a glossary of the evidence presented previously.

•

$(2)$ is tautological from Thomas-Yau semistability (cf. Remark 5.26).
•

$(1)$ implies Thomas-Yau semistability: see the Floer theoretic obstructions Thm. 3.21, Thm. 3.26, Conj. 3.31, where we justified this for immersed Lagrangians under the automatic transversality and the positivity condition, or alternatively by assuming Joyce’s program.
•

$(2)\iff(3)$ : this is a consequence of $(N,A)$ -potential clustering (cf. Cor. 5.9, section 5.2.1), the uniform bound for $\mathcal{S}-\bar{\mathcal{S}}$ (cf. section 5.2.2, 3.8.3), and the formula (58) for the elementary functional.
•

$(1)\implies(4)$ : see Prop. 3.40, where we justified this under automatic transversality and the positivity condition.
•

$(4)\implies(3)$ : obvious.

The rest of this section concerns $(2)\implies(4)$ , and the next section concerns $(4)\implies(1)$ . The arguments will rely on several unproven statements, which we consider plausible, but may involve rather significant difficulties or substantial foundational work. Nevertheless, we think it is instructive to see heuristically how everything fits together.

Conjecture 5.22.

In the semistable case, the Solomon functional has a minimizer.

Proof.

(Heuristic) First, we claim that for a minimizing sequence $L^{(k)}$ of the Solomon functional, without loss of generality the Lagrangian potential $f_{L}^{(k)}$ is a priori bounded:

\sup_{k}\left\lVert f_{L}^{(k)}\right\rVert_{L^{\infty}}\leq C.

(62)

Consider the potential clustering setup. We can adjust the Lagrangian potentials on $L_{i}$ by constants separately, and as long as $\sup_{L_{j}}f_{L_{j}}\leq\inf_{L_{i}}f_{L_{i}}$ for $j<i$ , this process will not affect the Novikov positivity requirement, so the Lagrangian branes should remain in $\mathcal{L}$ . We view $\sup_{L_{1}}f_{L_{1}},\sup_{L_{2}}f_{L_{2}}-\sup_{L_{1}}f_{L_{1}},\ldots,\sup_{L_{N}}f_{L_{N}}-\sup_{L_{N-1}}f_{L_{N-1}}$ as independent constants. Adjusting all potentials by a common constant does not affect the Solomon functional, but allows us to set $\sup_{L_{1}}f_{L_{1}}=0$ . Decreasing $\sup_{L_{i}}f_{L_{i}}-\sup_{L_{i-1}}f_{L_{i-1}}$ subject to the Novikov positivity requirement will decrease the elementary functional (58), crucially because of the semistability condition (61). The part $\mathcal{S}-\bar{\mathcal{S}}$ is unchanged. Thus after this adjustment, the sequence is still minimizing for the Solomon functional. We can thus achieve $\sup_{L_{i-1}}f_{L_{i-1}}=\inf_{L_{i}}f_{L_{i}}$ for all $i$ . By the potential clustering property, we then have (62).

Next we need the compactness from geometric measure theory. As discussed in section 5.1 and 5.2, under quantitative almost calibratedness there is an a priori volume bound, and the Lagrangians all remain in a fixed bounded subset of $X$ , so Federer-Fleming compactness (cf. Theorem 5.2) holds automatically. The uniform potential bound (62) would then justify that the weak limit is an almost calibrated Lagrangian current $L$ with bounded potential $f_{L}$ (cf. Lemma 5.7). The continuity of the Solomon functional (cf. Lemma 5.8) then shows $\mathcal{S}(L)=\inf_{\mathcal{L}}\mathcal{S}$ .

In section 5.3 we presented the evidence for the conjectural $L^{2}$ -smoothing property, which would allow us to assume a uniform a priori bound on the minimizing sequence

\int_{L}|\vec{H}|\leq C.

so we can use Allard compactness theorem 5.3. In effect, we can assume the minimizing sequence converges subsequentially both as currents and as varifolds. By assumption the class $\mathcal{L}$ is closed under the varifold/current topology of the Lagrangian, so the limit $L$ lies in $\mathcal{L}$ , whence provides a minimizer in $\mathcal{L}$ . ∎

Remark 5.27.

If we demand $\mathcal{L}$ is closed under the flat topology of currents, without requiring varifold convergence, then we would not need the difficult $L^{2}$ -smoothing property in the argument. However, this would allow the pathological behaviour in Example 5.4, which would increase the difficulty of Floer theory for weak regularity Lagrangians.

Remark 5.28.

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:

•

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.
•

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.
•

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.