We have emphasized that the Solomon functional depends not only on $L$, but also the potential $f_{L}$, and that the freedom of additive constants causes the space of $(L,f_{L})$ to be noncompact, even though the space of Lagrangians $\mathcal{L}$ is more or less compact under the varifold/current topology. We now wish to explain why the asymptotic behaviour of the Solomon functional should be controlled by Thomas-Yau semistability. The key tool is an a priori bound on the difference between the Solomon functional and the elementary functional, for which we gave sufficient conditions in section 3.8.3 and 5.2.2.

In the setup of $(N,A)$-potential clustering (cf. Cor. 5.9, section 5.2.1), we will rewrite the elementary functional $\bar{\mathcal{S}}$ (cf. (45)). Recall we have a Lagrangian $L$ built from $L_{1},\ldots L_{N}$; in the unobstructed immersed Lagrangian context, this structure comes from a twisted complex (cf. section 3.8.3). We introduce the new Lagrangian currents

$$\mathcal{E}_{k}=L_{1}+L_{2}\ldots+L_{k},\quad k=0,1,2,\ldots N,$$

which in the immersed context corresponds to the twisted complex (18). In particular $\mathcal{E}_{N}=L$, which is homologous to $L_{0}$. Thus

$$\begin{split}\bar{\mathcal{S}}=&\text{Im}\left(\sum_{1}^{N}(\sup_{L_{i}}f_{L_{i}})e^{-i\hat{\theta}}(\int_{\mathcal{E}_{i}}\Omega-\int_{\mathcal{E}_{i-1}}\Omega)\right)-(\sup_{L_{0}}f_{L_{0}})\text{Im}(e^{-i\hat{\theta}}\int_{L_{0}}\Omega)\\ =&\text{Im}\left(\sum_{1}^{N-1}(\sup_{L_{i}}f_{L_{i}}-\sup_{L_{i+1}}f_{L_{i+1}})e^{-i\hat{\theta}}\int_{\mathcal{E}_{i}}\Omega\right)+(\sup_{L_{N}}f_{L_{N}}-\sup_{L_{0}}f_{L_{0}})\text{Im}(e^{-i\hat{\theta}}\int_{L_{0}}\Omega).\end{split}$$

But we chose in the beginning $\hat{\theta}=\arg\int_{L_{0}}\Omega.$ Thus $\text{Im}(e^{-i\hat{\theta}}\int_{L_{0}}\Omega)=0$, and

$$\bar{\mathcal{S}}=\sum_{1}^{N-1}(\sup_{L_{i}}f_{L_{i}}-\sup_{L_{i+1}}f_{L_{i+1}})\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{E}_{i}}\Omega\right).$$

(58)

As part of the potential clustering property, we have

$$\sup_{L_{1}}f_{L_{1}}\leq\sup_{L_{2}}f_{L_{2}}\leq\ldots\leq\sup_{L_{N}}f_{L_{N}}.$$

(59)

We arrive at the following key dichotomy:

• •

  In the unstable case, there exists some $1\leq k\leq N-1$, such that

  $$\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{E}_{k}}\Omega\right)>0,$$

  or equivalently

  $$\arg\int_{\mathcal{E}_{k}}\Omega>\hat{\theta}.$$

  (60)

  Notice $L$ fits into a distinguished triangle

  $$\mathcal{E}_{k}\to L\to\cup_{i\geq k+1}L_{i}\to\mathcal{E}_{k}[1],$$

  We explained in Theorem 3.21 under the extra hypotheses of automatic transversality and the positivity condition, that this leads to a Floer theoretic obstruction. In Conjecture 3.31 we heuristically argued that even without these extra hypotheses, the Floer theoretic obstruction should follow from the Thomas-Yau-Joyce program.

  From a different perspective, we can add an arbitrarily large positive number $a>0$ to the Lagrangian potential on $L_{k+1},\ldots L_{N}$. This is compatible with the Novikov positivity condition, so $L$ stays unobstructed, but $\bar{S}$ changes by an unbounded amount

  $$-a\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{E}_{i}}\Omega\right)<0.$$

  We conclude that in the unstable case, the elementary functional is unbounded from below.

• •

  In the semistable case, for any $L$ in the class $\mathcal{L}$ that can be written in the twisted complex form as above, we always have

  $$\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{E}_{k}}\Omega\right)\leq 0,\quad\forall k=1,2,\ldots N-1.$$

  (61)

  Then the elementary functional (58) is nonnegative.

  In section 5.2 we argued that since the homology class of $L$ is prescribed a priori, subject to the quantitative almost calibrated assumption, only finitely many possibilities of homology classes can arise for $L_{i}$ in any decomposition. Thus the stronger condition

  $$\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{E}_{k}}\Omega\right)<0,\quad\forall k=1,2,\ldots N-1.$$

  would be equivalent to a uniform bound: for some small $c>0$,

  $$\text{Im}\left(e^{-i\hat{\theta}}\int_{\mathcal{E}_{k}}\Omega\right)\leq-c<0,\quad\forall k=1,2,\ldots N-1.$$

  This holds when the class $\mathcal{L}$ is stricly stable (cf. Definition 3.32). Together with potential clustering, it implies

  $$\bar{\mathcal{S}}(L)\geq c(\sup_{L}f_{L}-\inf_{L}f_{L}+A).$$

  Thus if the Lagrangian potential oscillation becomes unbounded, then the elementary functional goes to positive infinity. The geometric intuition is the properness of the Solomon functional modulo a global additive constant for $f_{L}$.

Since the Solomon functional and the elementary functional only differ by a bounded amount, the above conclusions transfer to the Solomon functional. Thus the Solomon functional is bounded below in the semistable case, and unbounded from below in the unstable case. A key slogan here is that the asymptotic behaviour of the Solomon functional is governed by Floer theory. This is analogous to the partially conjectural picture in the variational approach to the HYM equation, where the asymptotic behaviour of the Donaldson functional is governed by algebraic geometry (cf. section 2.5).