[proofplan]
We apply the direct method in the compactness theory of integral currents. First choose a minimizing sequence in the admissible class; the support constraint gives compactness in space, while the minimizing property and fixed boundary give uniform bounds on mass and boundary mass. The Federer-Fleming [compactness theorem](/theorems/2748) produces an integral-current limit supported in the same compact set, continuity of the boundary operator preserves the boundary condition, and lower semicontinuity of mass gives attainment of the infimum.
[/proofplan]
[step:Choose a minimizing sequence with uniformly bounded mass]
Let
\begin{align*}
\mu := \inf\{\mathsf{M}(R) : R \in \mathcal{A}_{\Gamma,K}(U)\}.
\end{align*}
Since $\mathcal{A}_{\Gamma,K}(U)$ is nonempty, choose $R_0 \in \mathcal{A}_{\Gamma,K}(U)$. Because $R_0 \in \mathcal{I}_m(U)$ and $\operatorname{spt} R_0 \subset K$ with $K$ compact, $\mathsf{M}(R_0) < \infty$, so $0 \leq \mu \leq \mathsf{M}(R_0) < \infty$.
For each $j \in \mathbb{N}$, by the definition of the infimum choose
\begin{align*}
T_j \in \mathcal{A}_{\Gamma,K}(U)
\end{align*}
such that
\begin{align*}
\mathsf{M}(T_j) \leq \mu + \frac{1}{j}.
\end{align*}
Then
\begin{align*}
\sup_{j \in \mathbb{N}} \mathsf{M}(T_j) \leq \mu + 1 < \infty.
\end{align*}
Moreover, since $\partial T_j = \Gamma$ for every $j \in \mathbb{N}$,
\begin{align*}
\sup_{j \in \mathbb{N}} \mathsf{M}(\partial T_j)
=
\mathsf{M}(\Gamma)
<
\infty.
\end{align*}
Finally, each $T_j$ satisfies $\operatorname{spt} T_j \subset K$ by membership in $\mathcal{A}_{\Gamma,K}(U)$.
[guided]
The direct method begins by replacing the abstract infimum with a sequence that nearly attains it. Define
\begin{align*}
\mu := \inf\{\mathsf{M}(R) : R \in \mathcal{A}_{\Gamma,K}(U)\}.
\end{align*}
The admissible class is assumed nonempty, so there is some current $R_0 \in \mathcal{A}_{\Gamma,K}(U)$. Since $R_0$ is an integral $m$-current in $U$ and its support is contained in the compact set $K \subset U$, its total mass is finite. Hence
\begin{align*}
0 \leq \mu \leq \mathsf{M}(R_0) < \infty.
\end{align*}
Now use the defining property of an infimum: for every $j \in \mathbb{N}$, there exists
\begin{align*}
T_j \in \mathcal{A}_{\Gamma,K}(U)
\end{align*}
with
\begin{align*}
\mathsf{M}(T_j) \leq \mu + \frac{1}{j}.
\end{align*}
This gives the uniform mass estimate
\begin{align*}
\sup_{j \in \mathbb{N}} \mathsf{M}(T_j) \leq \mu + 1 < \infty.
\end{align*}
The compactness theorem for integral currents also requires control of the boundary masses. Here the boundary is fixed: for every $j \in \mathbb{N}$, membership in $\mathcal{A}_{\Gamma,K}(U)$ gives $\partial T_j = \Gamma$. Therefore
\begin{align*}
\sup_{j \in \mathbb{N}} \mathsf{M}(\partial T_j)
=
\mathsf{M}(\Gamma)
<
\infty.
\end{align*}
The support hypothesis is also uniform: $\operatorname{spt} T_j \subset K$ for all $j \in \mathbb{N}$. Thus the sequence has exactly the three compactness inputs we need: bounded mass, bounded boundary mass, and support in one fixed compact subset of $U$.
[/guided]
[/step]
[step:Apply compactness to obtain an integral-current limit]
By the Federer-Fleming compactness theorem for integral currents with compact support and uniformly bounded mass and boundary mass, applied to the sequence $(T_j)_{j \in \mathbb{N}}$ in the [open set](/page/Open%20Set) $U$ with common compact support contained in $K$, there exist a subsequence $(T_{j_\ell})_{\ell \in \mathbb{N}}$ and an integral current
\begin{align*}
T \in \mathcal{I}_m(U)
\end{align*}
such that $T_{j_\ell} \to T$ weakly as currents in $U$.
(citing a result not yet in the wiki: Federer-Fleming Compactness Theorem for Integral Currents)
[/step]
[step:Show that the limit remains supported in $K$]
We prove $\operatorname{spt} T \subset K$. Let $\omega$ be any smooth compactly supported $m$-form on $U$ with $\operatorname{spt}\omega \subset U \setminus K$. Since $\operatorname{spt} T_{j_\ell} \subset K$, we have
\begin{align*}
T_{j_\ell}(\omega) = 0
\end{align*}
for every $\ell \in \mathbb{N}$. [Weak convergence](/page/Weak%20Convergence) of currents gives
\begin{align*}
T(\omega)
=
\lim_{\ell \to \infty} T_{j_\ell}(\omega)
=
0.
\end{align*}
Thus $T$ vanishes on every compactly supported smooth $m$-form supported in $U \setminus K$, which is precisely $\operatorname{spt} T \subset K$.
[/step]
[step:Pass the boundary condition to the limit]
We show that $\partial T = \Gamma$. Let $\eta$ be any smooth compactly supported $(m-1)$-form on $U$. By the definition of the boundary of a current,
\begin{align*}
(\partial T)(\eta) = T(d\eta).
\end{align*}
Since $T_{j_\ell} \to T$ weakly as currents and $d\eta$ is a smooth compactly supported $m$-form on $U$,
\begin{align*}
T(d\eta)
=
\lim_{\ell \to \infty} T_{j_\ell}(d\eta).
\end{align*}
Using again the definition of the boundary and the equality $\partial T_{j_\ell} = \Gamma$, we obtain
\begin{align*}
T(d\eta)
=
\lim_{\ell \to \infty} (\partial T_{j_\ell})(\eta)
=
\lim_{\ell \to \infty} \Gamma(\eta)
=
\Gamma(\eta).
\end{align*}
Therefore $(\partial T)(\eta) = \Gamma(\eta)$ for every smooth compactly supported $(m-1)$-form $\eta$ on $U$, and hence $\partial T = \Gamma$.
[/step]
[step:Use lower semicontinuity of mass to attain the infimum]
The mass functional is lower semicontinuous under weak convergence of integral currents with locally bounded mass. Applying this to $T_{j_\ell} \to T$ gives
\begin{align*}
\mathsf{M}(T)
\leq
\liminf_{\ell \to \infty} \mathsf{M}(T_{j_\ell}).
\end{align*}
By the minimizing property of the sequence,
\begin{align*}
\liminf_{\ell \to \infty} \mathsf{M}(T_{j_\ell})
\leq
\lim_{\ell \to \infty}\left(\mu + \frac{1}{j_\ell}\right)
=
\mu.
\end{align*}
Hence
\begin{align*}
\mathsf{M}(T) \leq \mu.
\end{align*}
(citing a result not yet in the wiki: Lower Semicontinuity of Mass for Integral Currents)
From the previous steps, $T \in \mathcal{I}_m(U)$, $\partial T = \Gamma$, and $\operatorname{spt} T \subset K$, so $T \in \mathcal{A}_{\Gamma,K}(U)$. By the definition of $\mu$ as the infimum over $\mathcal{A}_{\Gamma,K}(U)$,
\begin{align*}
\mu \leq \mathsf{M}(T).
\end{align*}
Combining the two inequalities gives
\begin{align*}
\mathsf{M}(T) = \mu
=
\inf\{\mathsf{M}(R) : R \in \mathcal{A}_{\Gamma,K}(U)\}.
\end{align*}
Thus $T$ is a mass-minimizing integral $m$-current in $U$ with boundary $\Gamma$ and support contained in $K$.
[/step]
