5.1 Compactness and regularity [04DT]

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5.1 Compactness and regularity

5.1.1 Standard geometric measure theory

We will use the standard language of geometric measure theory, see Federer [32] or Morgan [60] for the terminologies. The starting point of the variational approach is that there are foundational compactness theorems in geometric measure theory.

Theorem 5.2.

(Federer-Fleming compactness theorem [32]) Let $L_{i}$ be a sequence of $m$ -dimensional integral currents in a complete Riemannian manifold $X$ , all supported in a fixed bounded subset, with uniform bounds $\text{Mass}(L_{i})\leq C$ and $\text{Mass}(\partial L_{i})\leq C$ . Then up to subsequence $L_{i}$ converges weakly in the current topology to an $m$ -dimensional integral current $L$ with the same bounds.

Remark 5.2.

While compactness in the current topology is elementary, the claim that the limit is also an integral current is nontrivial, and can be viewed as a regularity result. The same holds with the Allard compactness theorem below. For our applications, we will always work with closed integral currents, namely $\partial L_{i}=0$ , which implies $\partial L=0$ in the limit. To such currents one can associate a homology class.

Remark 5.3.

A more technical version of Federer-Fleming compactness replaces the current topology by the flat norm topology, which is a slightly stronger topology. The flat norm of an integral current $T$ is

\left\lVert T\right\rVert_{flat}=\inf\{\text{Mass}(A)+\text{Mass}(B)|T=A+\partial B,\quad A,B\text{ are integral currents}\},

and the convergence $T_{i}\to T$ in this topology simply means $\left\lVert T-T_{i}\right\rVert_{flat}\to 0$ .

Theorem 5.3.

(Allard compactness [4]) Let $L_{i}$ be a sequence of $m$ -dimensional integer rectifiable varifolds in a complete Riemannian manifold $X$ , all supported in a fixed bounded subset, with a uniform volume upper bound $\text{Mass}(L_{i})\leq C$ and a uniform bound on the first variation $\int_{L_{i}}|\vec{H}|\leq C$ . Then up to subsequence, $L_{i}$ converges to an $m$ -dimensional integer rectifiable varifold $L$ with the same bounds.

Remark 5.4.

Federer-Fleming and Allard are somewhat complementary. Integral currents are a special kind of distribution valued forms, while varifolds are a special kind of measures on the real Grassmannian bundle $Gr(TX,m)$ over $X$ whose fibres parametrize $m$ -dimensional planes in the tangent spaces of $X$ . One key advantage of currents is that they know about orientations, while varifolds do not. The integral current $L$ recovers the underlying rectifiable subset $\text{supp}(L)$ with multiplicity, so can be canonically associated with a varifold $L^{var}$ . On the other hand, the natural topology on varifolds (i.e. the topology as measures on $Gr(TX,m)$ ) remembers tangent plane information, which can be lost under the flat norm convergence of integral currents. Morever, assuming all the varifolds in the sequence are contained in a bounded region, then the total volume mass converges under varifold convergence, but not necessarily so under flat norm convergence. The intuition is that morally the varifold topology detects one more derivative than the flat norm topology. This explains why Allard requires some integral control on the mean curvature, while Federer-Fleming does not.

We shall later use the informal terminology of ‘varifold/current topology’ to refer to convergence simultaneously in the varifold topology and the flat norm topology on integral currents.

Example 5.4.

Inside $S^{1}_{2\pi}\times\mathbb{R}$ with the standard Euclidean metric, take $L_{k}$ as the graph over $S^{1}$ of the function $\frac{1}{k}\sin(kx)$ . Then $L_{k}$ are Lagrangian currents, which converge to $S^{1}$ as currents, but due to the high oscillation, $\liminf Mass(L_{k})>Mass(S^{1})$ , and $L_{k}$ do not converge to $S^{1}$ in the varifold sense. The Lagrangian angle of $L_{k}$ is prescribed by $\tan\theta=\cos(kx)$ , which converges to zero in the current sense, but not strongly in $L^{1}$ .

One of the best regularity theorems in geometric measure theory is

Theorem 5.5.

(Almgren’s big regularity theorem [5]) Let $L$ be a compactly supported $m$ -dimensional closed integral current inside a complete Riemannian manifold, which minimizes the volume among all closed integral currents in the same homology class, then away from a closed subset $S\subset\text{supp}(L)$ with Hausdorff dimension at most $m-2$ , the rectifiable subset $\text{supp}(L)\setminus S$ is a smooth submanifold.

Remark 5.5.

Real codimension two singularity is the optimal result, as easily seen from the examples of singular algebraic curves in $\mathbb{CP}^{2}$ , which are automatically area minimizers in their homology classes.

Remark 5.6.

Almgren’s big regularity theorem is well known for its monumental size of around 1000 pages. The recent works of Delellis et al. have somewhat simplified the proof, which still remains very nontrivial (cf. [24] for some introduction).

A standard way to apply these theorems, for instance inside a compact ambient space, is to fix the homology class, and minimize the volume among all the integral currents therein. The compactness theorem guarantees the existence of an absolute volume minimizer, and the regularity theorem then improves its regularity to be more like submanifolds. This strategy is highly effective in producing minimal surfaces, but there is no useful criterion⁵⁹⁵⁹ 59 If there is at least one special Lagrangian within the given homology class, then all absolute minimizers must be special Lagrangians, by an easy calibration argument. This however does not answer how to find the special Lagrangian in the first place. to guarantee the volume minimizers to be special Lagrangians, which is why producing special Lagrangians is a highly nontrivial problem in geometric measure theory.

5.1.2 Exact Lagrangians under weak regularity

We need to ensure the class of Lagrangians in the variational setup is closed under the varifold/current topology. A trivial observation is

Lemma 5.6.

Let $L_{i}$ be closed Lagrangian integral currents, and suppose $L_{i}\to L$ in the current topology, then the Lagrangian/quantitative almost calibratedness conditions pass to the limit.

Let $L$ be a closed Lagrangian integral current, and $f_{L}$ be an $L^{\infty}$ function on $L$ . We say the exact condition $\lambda=df_{L}$ holds in the weak sense, if for any compactly supported test $(n-1)$ -form $\chi$ ,

\int_{L}\lambda\wedge\chi=-\int_{L}f_{L}d\chi.

(54)

To make sense of the RHS, notice the rectifiability of $L$ allows the integration of the $L^{\infty}$ -valued $n$ -form $f_{L}d\chi$ . Equivalently, the normal current $f_{L}L$ has distributional derivative $\chi\to\int_{L}\chi\wedge\lambda$ .

Remark 5.7.

The examples of immersed Lagrangians show that we cannot require $f_{L}$ to have a continuous extension to $X$ , so $L^{\infty}$ -regularity is the best we can impose on $f_{L}$ .

Lemma 5.7.

All Lagrangians are assumed to be contained in a fixed bounded region of $X$ , homologous to $L_{0}$ , and are quantitatively almost calibrated. If $L_{i}$ is a sequence of exact Lagrangians with potential $f_{L_{i}}$ , such that $f_{L_{i}}$ are uniformly bounded in $L^{\infty}$ . Then up to subsequence, there is a Lagrangian $L$ with potential $f_{L}$ , such that $L_{i}\to L$ and $f_{L_{i}}L_{i}\to f_{L}L$ as currents.

Proof.

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 $f_{L}$ . ∎

Continuity of the Solomon functional

Inside Stein manifolds, the Solomon functional can be defined for any Lagrangian $L$ with potential which is homologous to $L_{0}$ , without further Floer theoretic inputs: the formula (20) makes sense after choosing any bordism current $\mathcal{C}$ with $\partial\mathcal{C}=L-L_{0}$ in the sense of currents, and the choice does not matter.

Lemma 5.8.

(Continuity of the Solomon functional) All Lagrangians are assumed to be contained in a fixed bounded region of $X$ , homologous to $L_{0}$ , and are quantitatively almost calibrated. Suppse $L_{i}$ is a sequence of Lagrangian integral currents with potential $f_{L_{i}}$ , such that $L_{i}\to L$ in the flat norm, and $f_{L_{i}}L_{i}$ converge to $f_{L}L$ as currents, then the Solomon functionals converge: $\mathcal{S}(L_{i})\to\mathcal{S}(L)$ .

Proof.

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.

∎

Robustness of potential clustering

We revisit the potential clustering property (cf. section 3.8.3) from the geometric measure theory perspective. Assume as always that the Lagrangian integral current $L$ is quantitatively almost calibrated, homologous to $L_{0}$ , and equipped with Lagrangian potential $f_{L}$ . Given constants $N\in\mathbb{N},A\in\mathbb{R}_{+}$ , we say $L$ satisfies $(N,A)$ -potential clustering, if

L=\sum_{1}^{N}L_{i},\quad f_{L}L=\sum_{1}^{N}f_{L_{i}}L_{i},

for quantitatively almost calibrated, closed Lagrangian integral currents $L_{i}$ with potential $f_{L_{i}}$ , contained inside the support of $L$ , such that the oscillation of the Lagrangian potentials have uniform bounds

\sup_{L_{i}}f_{L_{i}}-\inf_{L_{i}}f_{L_{i}}\leq A,

while for any $i>j$ ,

\sup_{L_{j}}f_{L_{j}}\leq\inf_{L_{i}}f_{L_{i}}.

Without loss of generality, we assume $\sup_{L_{0}}f_{L_{0}}-\inf_{L_{0}}f_{L_{0}}\leq A$ for the fixed Lagrangian $L_{0}$ .

Remark 5.8.

Here we allow $L_{i}$ to have overlapping supports. For instance, it is possible for $L_{1}=L_{2}$ as currents, but $f_{L_{2}}$ and $f_{L_{1}}$ differ by a constant.

We will later be interested in uniform upper bounds on $N,A$ . For now, we observe the robustness under limits:

Corollary 5.9.

Fix the choice of $N,A$ . Suppose we are given a sequence of Lagrangians $L^{(j)}$ with potential satisfying $(N,A)$ -potential clustering, and assume $L_{i}^{(j)}\to L_{i}$ as currents for $1\leq i\leq N$ , all $f_{L_{i}}^{(j)}$ have uniform $L^{\infty}$ bounds, and $f_{L_{i}^{(j)}}L^{(j)}\to f_{L_{i}}L_{i}$ as currents. Then the limit $L=\sum_{1}^{N}L_{i}$ with its potential $f_{L}$ also satisfies $(N,A)$ -potential clustering.

Proof.

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. ∎

5.1.3 Almgren’s regularity in the almost Calabi-Yau setting?

The main reason to impose the Calabi-Yau condition is so that any special Lagrangian closed integral current is an absolute volume minimizer among all closed integral currents in the same homology class, by the calibration inequality (2). If we believe Thomas-Yau conjecture to be valid more generally for almost Calabi-Yau ambient structures, with

\frac{\omega^{n}}{n!}=(-1)^{n(n-1)}(\frac{\sqrt{-1}}{2})^{n}e^{2\rho}\Omega\wedge\overline{\Omega},

then we are naturally motivated to ask if Almgren’s regularity extends to special Lagrangians in this setting:

Question 11.

Suppose $L$ is a compactly supported closed integral current inside an almost Calabi-Yau manifold, which is a special Lagrangian in the sense of (1). Does it imply the support of $L$ is smooth away from a Hausdorff codimension two subset?

The following observations, left as easy exercises, are indications that almost Calabi-Yau manifolds behave similarly as Calabi-Yau manifolds.

•

By a variant of the calibration inequality (2), special Lagrangians minimize the weighted volume

$\int_{L}e^{-\rho}dvol_{L}\geq|\int_{L}\Omega|$

within its homology class.
•

Under the smoothness assumption, the mean curvature of a Lagrangian submanifold with phase function $\theta$ satisfies the formula

$\vec{H}=-\nabla^{\perp}\rho+J\nabla\theta,$

where $\nabla^{\perp}$ means the normal projection of the gradient, and $\nabla\theta$ is the derivative of $\theta$ along $L$ . Thus for smooth special Lagrangians in a bounded region, we have the a priori bound $|\vec{H}|\leq C$ .