Compactness Theorem

Inside a Ball $B$ in $\Bbb{R}^3$,

$$\{{\rm rectifiable\ currents\ }S{\rm\ in\ }BL {\rm\ \href{Area}\ }S\leq c, {\rm\ length\ }\partial S\leq c\}$$

is compact under the Flat Norm.

References

Morgan, F. ``What Is a Surface?'' Amer. Math. Monthly 103, 369-376, 1996.