(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.