5.1.1 Standard geometric measure theory [04DU]

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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.