Allard compactness requires an a priori bound $\int_{L}|\vec{H}|\leq C$, which cannot be implied by the quantitative almost calibrated condition, since the mean curvature involves one more derivative than the Lagrangian angle. However, for the purpose of our variational strategy, it is enough to ensure any minimization sequence of $\mathcal{S}$ can be replaced by a sequence with $\int_{L_{i}}|\vec{H}|\leq C$.

We think the smoothing property may be quite deep, and our limited attempt here is to explain how it relates to Joyce’s LMCF program, which suggests the smoothing property may hold with $p=2$. Recall the defining feature of the Solomon functional is its variation property under exact isotopies among unobstructed objects:

$$\delta\mathcal{S}=\int_{L}h\text{Im}(e^{-i\hat{\theta}}\Omega),$$

which holds under sufficient smoothness assumptions. Under a sufficiently smooth LMCF $(L_{t})$ in a Calabi-Yau manifold, the Lagrangians evolve by the local Hamiltonian function $-\theta_{t}$ up to an inconsequential additive constant (cf. section 4.1), so $\mathcal{S}$ evolves by

$$\begin{split}&\partial_{t}\mathcal{S}=-\int_{L_{t}}(\theta_{t}-\hat{\theta})\text{Im}(e^{-i\hat{\theta}}\Omega)=-\int_{L_{t}}(\theta_{t}-\hat{\theta})\text{Im}(e^{i(\theta-\hat{\theta})})dvol_{L_{t}}\\ =&-\int_{L_{t}}(\theta_{t}-\hat{\theta})\sin(\theta_{t}-\hat{\theta})dvol_{L_{t}}.\end{split}$$

(57)

If $L_{t}$ is almost calibrated, then $-\pi<\theta-\hat{\theta}<\pi$, so $\partial_{t}\mathcal{S}\leq 0$. We conclude that the Solomon functional decreases in time along Joyce’s LMCF under the almost calibrated assumption, at least for the time between the surgeries. It is plausible $\mathcal{S}$ is either continuous or jumps downwards at the surgeries in Joyce’s LMCF,⁶¹⁶¹ 61 A somewhat analogous phenomenon in the Brakke flow is that the total volume mass is either continuous or can only jump downwards in time. The mass loss is typically related to the disappearance of a component of the evolving varifold, which is conceptually similar to ‘collapsing zero objects’ in Joyce’s LMCF. This is ruled out by the almost calibrated condition, so optimistically one can even hope for the continuity of the Solomon functional in the almost calibrated setting. which would then imply the Solomon functional is monotone decreasing for all time.

Now recall that the heat equation on the Lagrangian angle implies an integral bound on the mean curvature (53). In particular, if the LMCF can be run for a definite amount of time $T$, then there exists some $t\leq T$, with

$$\int_{L_{t}}|\vec{H}|^{2}dvol_{L_{t}}\leq T^{-1}\int_{L_{t=0}}\theta^{2}dvol_{L_{t=0}}\leq CT^{-1},$$

where crucially the a priori constant $C$ does not depend on any quantitative smoothness assumption on the initial Lagrangian, provided it is quantitatively almost calibrated. Such $L_{t}$ would be a good candidate for $L^{\prime}$, subject to the hypothesis that Joyce’s LMCF remains within the class of Lagrangians $\mathcal{L}$.

Morally the class $\mathcal{L}$ arises as varifold/current limits of those Lagrangians admissible in Joyce’s program. Under the plausible assumption that Joyce’s LMCF can be passed to the varifold/current limit, then the $L^{2}$-smoothing property can be well explained. The $\mathcal{S}(L^{\prime})\leq\mathcal{S}(L)$ condition comes from the decrease of the Solomon functional along the flow, and the $\int_{L^{\prime}}|\vec{H}|^{2}dvol_{L^{\prime}}\leq C$ condition would follow if Joyce’s LMCF can be run for a uniform amount of time $T$. If $T$ can be taken arbitrarily large, then we can demand further that the $L^{2}$ mean curvature is arbitrarily small.