Calculus of variations and optimal control; optimization Geometric measure theory and optimization on manifolds Theorem

Federer–Fleming compactness theorem

Builds on Integral current Flat norm Current Area-minimizing current

Statement

Let $0 \le m \le n$ , let $K \subseteq \R^n$ be compact, and let $C > 0$ . Let $(T_j)_{j \ge 1}$ be a sequence of integral currents of dimension $m$ on $\R^n$ with support in $K$ . Assume $\mathbf{M}(T_j) + \mathbf{M}(\partial T_j) \le C$ for all $j$ , where $\mathbf{M}$ is the mass and $\partial T_j = 0$ when $m = 0$ . Then there are a subsequence $(T_{j_i})$ and an integral $m$ -current $T$ with support in $K$ such that $\mathbf{F}(T_{j_i} - T) \to 0$ , where $\mathbf{F}$ is the flat norm. Moreover $\mathbf{M}(T) \le \liminf_i \mathbf{M}(T_{j_i})$ and $\mathbf{M}(\partial T) \le \liminf_i \mathbf{M}(\partial T_{j_i})$ .

Remarks

Weak convergence of a subsequence follows from the mass bound by a diagonal argument on a countable dense set of forms. The mass is lower semicontinuous under weak convergence, because it is a supremum of the continuous functionals $T \mapsto T(\omega)$ . The hard part is the closure statement: the weak limit is again integral, and the convergence holds in the flat norm. Federer and Fleming proved it with the deformation theorem, which pushes currents onto a cubical grid; later proofs use slicing.

Example at work: let $B$ be an integral $(m-1)$ -current with compact support that is the boundary of some integral $m$ -current. A minimising sequence for the mass among integral currents with boundary $B$ can be projected into a fixed ball without increasing mass, and its boundary masses are all $\mathbf{M}(B)$ . The theorem gives a limit with boundary $B$ and least mass, an area-minimizing current.

The bound on $\mathbf{M}(\partial T_j)$ is needed. In $\R$ , let $T_j = \sum_{i=0}^{j-1} [\![i/j, \, i/j + 1/(2j)]\!]$ , integral $1$ -currents in $[0, 1]$ with mass $\frac{1}{2}$ and $\mathbf{M}(\partial T_j) = 2j$ . They converge weakly to $\frac{1}{2} [\![0, 1]\!]$ , which has multiplicity $\frac{1}{2}$ and is not an integral current.

Going deeper

For a student

The theorem is the compactness step of the direct method of the calculus of variations for currents. Combined with lower semicontinuity of mass, it solves the Plateau problem for integral currents in every dimension and codimension. Regularity theory, such as the results of De Giorgi, Almgren and Allard, then shows that the minimisers are smooth away from a small singular set. The same compactness, with the flat norm, underlies the theory of flat chains and of minimal surfaces obtained by min-max methods.

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Area-minimizing current concept
Current concept
Flat norm concept
Integral current concept
Federer–Fleming compactness theorem statement

Tag	`TPXYARH912`
URI	`m2://TPXYARH912`
Kind	statement, theorem
Writing	W0 stub
MSC	49Q15, 28A75, 49Q20
Status	This copy of the site has no build data. Run `m2 status TPXYARH912`.
About	Integral current
Uses	Integral current, Flat norm, Current, Area-minimizing current