By Lemma 2.1 the volume mass is uniformly upper bounded. By Federer-Fleming compactness, subsequentially $L_{i}\to L$ in the flat topology for some Lagrangian integral current $L$ homologous to $L_{0}$. This implies $\int_{L_{k}}g\text{Re}\Omega\to\int_{L}g\text{Re}\Omega$ for any $C^{\infty}$ test function $g$, even though $\liminf_{i}Mass(L_{i})$ may be strictly greater than $Mass(L)$, as we do not assume varifold convergence.

We focus on a coordinate ball. The $n$-currents $f_{L_{k}}L_{k}$ can be viewed as a collection of ${2n\choose n}$ signed measures $g\mapsto\int_{L_{k}}f_{L_{k}}gdx_{i_{1}}\wedge\ldots dx_{i_{n}}$. Each of these measures are bounded by the measure

$$g\mapsto(\sup\left\lVert f_{L_{i}}\right\rVert_{L^{\infty}})\frac{C}{\sin\epsilon}\int_{L_{k}}g\text{Re}\Omega,$$

whose total mass is uniformly bounded for all $k$. By the weak compactness of measures, subsequentially these signed measures converge, and the limiting signed measures have $L^{\infty}$ Radon-Nykodim derivatives $a_{i_{1}\ldots i_{n}}(x)$ with respect to the measure $g\mapsto\int_{L}g\text{Re}\Omega$:

$$\int_{L_{k}}f_{L_{k}}gdx_{i_{1}}\wedge\ldots dx_{i_{n}}\to\int_{L}ga_{i_{1}\ldots i_{n}}\text{Re}\Omega.$$

Thus inside the coordinate ball, the currents $f_{L_{k}}L_{k}$ converge to $\lim_{k}f_{L_{k}}L_{k}$:

$$\eta\mapsto\int_{L}\sum_{i_{1}<i_{2}\ldots<i_{n}}\eta(\partial_{i_{1}}\wedge\ldots\partial_{i_{n}})a_{i_{1}\ldots i_{n}}\text{Re}\Omega,$$

where $\eta$ is any test $n$-form.

Now $L$ is an integral current, so $\mathcal{H}^{n}$-a.e. $y\in\text{supp}(L)$ there is a well defined tangent space $T_{y}L$ and a local integer multiplicity $\Theta(y)$. Recall a blow up limit of an $n$-current $N$ at a point $y\in X$ refers to a subsequential limit of the currents on $T_{y}X$ as $r\to 0$:

$$\eta\mapsto\int_{N}\text{rescale}_{y,r}^{*}\eta,\quad\text{rescale}_{y,r}:x\mapsto\frac{x}{r}\text{ in the geodesic coordinates around $y$}.$$

For a.e $y\in\text{supp}(L)$, there is a unique blow up limit for the current $\lim_{k}f_{L_{k}}L_{k}$, which is

$$\eta\mapsto\int_{T_{y}L}\sum\eta(\partial_{i_{1}}\wedge\ldots\partial_{i_{n}})a_{i_{1}\ldots i_{n}}(y)\Theta(y)\text{Re}\Omega,$$

whose ${2n\choose n}$ component signed measures are just constant multiples of the Lebesgue measure on $T_{y}X$.

Observe that the weak formulation (54) passes to the limit:

$$\int_{L}\lambda\wedge\chi=-(\lim_{k}f_{L_{k}}L_{k})(d\chi).$$

Thus the blow up limit of $\lim_{k}f_{L_{k}}L_{k}$ at a.e. $y\in\text{supp}(L)$ is in fact a closed current. Consequently, the polyvector

$$\sum_{i_{1}<i_{2}\ldots<i_{n}}a_{i_{1}\ldots i_{n}}\partial_{i_{1}}\wedge\ldots\partial_{i_{n}}$$

must be a pure tensor lying in $\Lambda^{n}T_{y}L\subset\Lambda^{n}T_{y}X$. Hence

$$\lim_{k}f_{L_{k}}L_{k}=f_{L}L$$

for some $L^{\infty}$-function $f_{L}$. ∎

Since $f_{L_{i}}L_{i}$ converges to $f_{L}L$ as currents,

$$\int_{L_{i}}f_{L_{i}}\Omega\to\int_{L}f_{L}\Omega.$$

It suffices to justify $\int_{\mathcal{C}_{i}}\lambda\wedge\Omega\to\int_{\mathcal{C}}\lambda\wedge\Omega,$ where $\partial\mathcal{C}_{i}=L_{i}-L_{0}$, and $\partial\mathcal{C}=L-L_{0}$.

Now the flat norm convergence gives $L_{i}-L=A_{i}+\partial B_{i}$ for some integral currents $A_{i},B_{i}$, with $Mass(A_{i})+Mass(B_{i})\to 0$. Since $[L_{i}]=[L]=[L_{0}]\in H_{n}(X)$, the homology class of $A_{i}$ is zero. A version of the isoperimetric theorem (cf. Prop. 5.11 below) then says $A_{i}=\partial B_{i}^{\prime}$ with $Mass(B_{i}^{\prime})\leq CMass(A_{i})^{(n+1)/n}\to 0$. Without loss of generality we absorb $B_{i}^{\prime}$ into $B_{i}$. Then we can simply choose $\mathcal{C}_{i}=\mathcal{C}+B_{i}$, which is legitimate since it satisfies $\partial\mathcal{C}_{i}=L_{i}-L_{0}$. The claim follows by

$$|\int_{B_{i}}\lambda\wedge\Omega|\leq CMass(B_{i})\to 0.$$

∎

Notice a potential bound such as $f_{L}\geq c$ can be characterized by the positivity of the measure $g\mapsto\int_{L}(f_{L}-c)g\text{Re}\Omega$. This characterization is robust under current convergence, so

$$\sup_{L_{i}}f_{L_{i}}\leq\limsup_{j}\sup_{L_{i}^{(j)}}f_{L_{i}^{(j)}},\quad\inf_{L_{i}}f_{L_{i}}\geq\liminf_{j}\inf_{L_{i}^{(j)}}f_{L_{i}^{(j)}},$$

hence the potential clustering bounds pass to the limit. ∎

The isoperimetric theorem [71, Thm 6.1] guarantees the existence of an integral current $A^{\prime}$ supported in $B_{R}$ such that $\partial A^{\prime}=\partial A$ and for which

$$Mass(A^{\prime})^{(n-1)/n}\leq CMass(\partial A).$$

Notice the metric uniform equivalence means we do not need to be careful to distinguish the Hausdorff measure for the Euclidean metric and the Calabi-Yau metric. Let $T$ denote the cone over the current $A-A^{\prime}$, then $\partial T=A-A^{\prime}$, and thus by the quantitative calibrated condition,

$$\begin{split}&Mass(A)\leq\frac{1}{\sin\epsilon}\int_{A}\text{Re}\Omega=\frac{1}{\sin\epsilon}\int_{A^{\prime}}\text{Re}\Omega+\frac{1}{\sin\epsilon}\int_{\partial T}\text{Re}\Omega\\ \leq&\frac{1}{\sin\epsilon}Mass(A^{\prime})+\frac{1}{\sin\epsilon}\int_{T}d\text{Re}\Omega\leq\frac{1}{\sin\epsilon}CMass(\partial A)^{n/(n-1)},\end{split}$$

which is the isoperimetric inequality. ∎

We first isometrically embed $X$ into an ambient Euclidean space $\mathbb{R}^{N}$, so $A$ can be regarded as an integral current compactly supported in $X$. Fix a small number $\rho_{0}>0$ such that over $U$ the $\rho_{0}$-neighbourhood in $\mathbb{R}^{N}$ is isomorphic to the normal bundle, so there is a smooth retraction map $F$ back to $U$. The Lipschitz norm of $F$ is approximately one.

Applying the deformation theorem for $\mathbb{R}^{N}$ [71, section 5.3] to the current $\partial A$, with a parameter $\rho\leq\rho_{0}$ to be fixed, we can write

$$\partial A=P+\partial R,$$

where $P,R$ are integral currents inside $\mathbb{R}^{N}$, supported in the $O(\rho)$ neighbourhood of $\text{supp}(\partial A)$, with

$$Mass(R)\leq C\rho Mass(\partial A),\quad Mass(P)\leq CMass(\partial A),$$

where the constant $C$ depends only on $N,m$. Morever, $P$ is an integral linear sum of $m$-dimensional faces in the standard grid decomposition of $\mathbb{R}^{N}$ with cube size $\rho$. We now push forward via $F$:

$$F_{*}P+\partial F_{*}R=F_{*}(\partial A)=\partial A,$$

since $\partial A\subset U\subset X$ is fixed by $F$. Note that $F_{*}P,F_{*}R$ both live inside $U$, and their mass bounds are essentially the same as $P,R$ respectively.

Suppose first that $Mass(\partial A)\ll 1$. If $P$ is nonzero, then by the grid description of $P$,

$$\rho^{m}\leq Mass(P)\leq CMass(\partial A)$$

So by choosing $\rho=2(CMass(\partial A))^{1/m}$ in the above, we force $P=0$, so $\partial A=\partial F_{*}R$, with mass bound $Mass(F_{*}R)\leq\text{const}Mass(\partial A)^{(m+1)/m}$, so it suffices to take $A^{\prime}=F_{*}R$.

Now suppose $Mass(\partial A)\gtrsim 1$, then we choose $\rho=\rho_{0}$. Without loss of generality, we can replace $\partial A$ by $F_{*}P$, and pretend $R=0$. We know

• •

  $P$ is an integral linear combination of grid cube faces, where all the cubes lie in a bounded region of a fixed grid,

• •

  $F_{*}P$ is an exact current on $X$.

The set of all such $P$ form a finitely generated abelian group, which by classification is isomorphic to the direct sum of $\oplus_{1}^{r}\mathbb{Z}e_{i}$ and a finite abelian group. For any given element

$$P=\sum a_{i}e_{i}+\text{finite group part},$$

the linear coefficients $a_{i}$ of $e_{i}$ for $i=1,\ldots r$ are bounded by $|a_{i}|\lesssim Mass(P)\lesssim Mass(\partial A)$. Each $e_{i}$ gives rise to an exact simplicial chain $F_{*}e_{i}$ inside $X$, which is the boundary of a finite mass integral current $Q_{i}$. Thus

$$Mass(\sum_{1}^{r}a_{i}Q_{i})\leq\sum_{1}^{r}|a_{i}|Mass(Q_{i})\leq\text{const}\cdot Mass(\partial A).$$

The finite group part gives rise to another simplicical chain inside $X$ which is the boundary of some finite mass integral current. Thus we have produced an integral current $A^{\prime}$ with $\partial A^{\prime}=\partial A$, and mass bound

$$Mass(A^{\prime})\leq C(Mass(\partial A))+C\leq\text{const }Mass(\partial A).$$

since we are in the $Mass(\partial A)\gtrsim 1$ case. ∎

Let $f(r)=\mathcal{H}^{n}(L\cap B(r))>0$, then $f$ is increasing in $r$, and for a.e. $0<r<R$, by the coarea formula,

$$f^{\prime}(r)=\int_{\partial B(r)\cap L}\frac{1}{|\nabla r|}d\mathcal{H}^{n-1}\geq C^{-1}\mathcal{H}^{n-1}(\partial B(r)\cap L)\geq C^{-1}f(r)^{(n-1)/n}.$$

The last inequality is the isoperimetric inequality. Thus $\frac{d}{dr}f^{1/n}\geq C^{-1},$ whence we have the volume lower bound $f(r)^{1/n}\geq C^{-1}r$. ∎

From Lemma 2.1 there is an a priori volume bound $\text{Vol}(L)\leq C$. But if $P$ is in the support of $L$, then the regularity scale of $X$ near $P$ is bounded by

$$R^{n}\leq C\text{Vol}(L\cap B(R))\leq C,$$

whence $P$ must remain in a fixed compact subset. ∎

The asymptotic hypothesis on the regularity scale implies in particular that the regularity scale has a global lower bound on $X$. This gives a uniform lower bound on $\text{Vol}(L)$, which by the quantitative almost calibratedness gives a lower bound on the homological integral $\int_{L}\text{Re}\Omega$. ∎

Let $d=dist(P,P^{\prime})\geq N=[d]$ be the intrinsic diameter of $L$. By connectedness, we can find $P_{1},\ldots P_{N}$ with $dist(P,P_{i})=i$. Now the intrinsic balls $\hat{B}(P_{i},\frac{1}{2})$ are disjoint, but each takes up a nontrivial amount of volume $\geq C^{-1}$. Thus

$$C^{-1}N\leq\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re}(\Omega)\leq C,$$

so there is an a priori bound on $N$, hence on $d$. ∎

Given any $P,P^{\prime}$ on $L$, we write the potential as a line integral of the Liouville 1-form

$$f_{L}(P)-f_{L}(P^{\prime})=\int_{P^{\prime}}^{P}\lambda\leq\left\lVert\lambda\right\rVert_{C^{0}}dist(P,P^{\prime}).$$

Thus the intrinsic ball volume lower bound implies that if $a$ lies in the range of $f_{L}$, then

$$\text{Vol}(\{|f_{L}-a|<\frac{1}{2}\})\geq C^{-1}.$$

The range of $f_{L}$ is by assumption a closed interval. If the interval has length $\geq N$, then we can find $N$ distinct values of $a$ with disjoint $\{|f_{L}-a|<\frac{1}{2}\}$, so

$$C^{-1}N\leq\text{Vol}(L)\leq\frac{1}{\sin\epsilon}\int_{L}\text{Re}(\Omega)\leq C,$$

This provides an a priori bound on $N$, hence on the potential oscillation. ∎

By Lemma 6.3, a general immersed Lagrangian can be decomposed into a union of Lagrangians $L_{i}$ such that the ranges of the potentials $f_{L_{i}}$ are connected, and any bounding cochain structure naturally produces a twisted complex. The oscillation of each $L_{i}$ is bounded in terms of the quantitative almost calibration condition, and the ambient features of $X$. Morever, the number of Lagrangian components is also a priori bounded. ∎

We know $L-L_{0}$ is homologous to zero in $X$, and contained in a bounded subset of $X$ by Cor. 5.13. By a version of the isoperimetric theorem (cf. Prop. 5.11), we can find some compactly supported integral current $\mathcal{C}$ with $\partial\mathcal{C}=L-L_{0}$, with mass bound

$$\text{Mass}(\mathcal{C})\leq\text{const}\cdot\min\{\text{Mass}(L-L_{0})^{(n+1)/n},\text{Mass}(L-L_{0})\}.$$

This $\mathcal{C}$ has no relation to holomorphic curves. The quantitative almost calibratedness implies a volume bound on $L$ (cf. Lemma 2.1), hence $\text{Mass}(\mathcal{C})$ is a priori bounded. The homological nature of the Solomon functional (20) gives

$$\mathcal{S}(L)=\int_{L}f_{L}\text{Im}(e^{-i\hat{\theta}}\Omega)-\int_{L_{0}}f_{L_{0}}\text{Im}(e^{-i\hat{\theta}}\Omega)-\text{Im}\int_{\mathcal{C}}\lambda\wedge e^{-i\hat{\theta}}\Omega.$$

The mass bound then implies

$$|\int_{\mathcal{C}}\lambda\wedge e^{-i\hat{\theta}}\Omega|\leq C\left\lVert\lambda\right\rVert_{C^{0}}\left\lVert\Omega\right\rVert_{C^{0}}\text{Mass}(\mathcal{C})\leq\text{const}.$$

Here since $L$ is contained in a bounded region, the terms $\lambda$ and $\Omega$ are bounded. Finally, using the potential clustering bound,

$$\begin{split}&|\int_{L}f_{L}\text{Im}(e^{-i\hat{\theta}}\Omega)-\int_{L_{0}}f_{L_{0}}\text{Im}(e^{-i\hat{\theta}}\Omega)-\bar{\mathcal{S}}(L)|\\ &\leq A(\int_{L}|\text{Im}(e^{-i\hat{\theta}}\Omega)|+\int_{L_{0}}|\text{Im}(e^{-i\hat{\theta}}\Omega)|)\\ &\leq A(\text{Mass}(L)+\text{Mass}(L_{0}))\leq\frac{2A}{\sin\epsilon}\int_{L_{0}}\text{Re}\Omega.\end{split}$$

Combining the above shows the a priori bound on $|\mathcal{S}(L)-\bar{\mathcal{S}}(L)|$. ∎