Behaviour of mass for currents with disjoint supports I am sorry if this is a basic question, but I don't think in MSE I will receive any answers.
Let $(M^3,g)$ be a compact and oriented Riemannian $3$-manifold. Let $\alpha$ and $\beta$ be the integral currents supported on the compact, connected, oriented embedded surfaces $\Sigma_1$ and $\Sigma_2$ with multiplicities $\theta_1$ and $\theta_2$, that is:
$$ \alpha(\omega) = \int_{\Sigma_1} \langle \omega(x), \tau_1 (x) \rangle \theta_1(x) d \mathcal{H}^2(x), \quad \omega \in \Omega_2(M),  $$
and similarly for $\beta$. Here, $\tau_1$ is a $2$-plane field that orients $\Sigma_1$ and $\theta_1$ is an integrable positive integer-valued function on $\Sigma_1$.
Assume that $\Sigma_1$ and $\Sigma_2$ are disjoint. Is it true that the mass of $\alpha$ is smaller than the mass of the sum $\alpha + \beta$?
 A: Yes, this is true, from the fact that the supports of these currents are disjoint. In fact, I think the same argument should work for two general currents, as long as their supports have disjoint neighborhoods.
Write $M(\alpha)$ and $M(\alpha + \beta)$ for the masses of $\alpha$ and $\alpha + \beta$, respectively.
Then there is a sequence of 2-forms $\omega_k$ such that $|\omega_k|_{C^0(M)} = 1$ for every $k$ and
$$|\alpha(\omega_k)| \to M(\alpha).$$
Let $\chi: M \to \mathbb{R}$ be a smooth, non-negative cutoff function that is equal to $1$ in a neighborhood of $\Sigma_1$ and $0$ in a neighborhood of $\Sigma_2$. Then it follows that
$$\alpha(\omega_k) = \alpha(\chi\omega_k)$$
for every $k$, and moreover
$$\beta(\chi\omega_k) = 0$$
for every $k$.
It follows that
$$|(\alpha + \beta)(\chi\omega_k)| \to M(\alpha).$$
However, by definition,
$$|(\alpha + \beta)(\chi\omega_k)| \leq M(\alpha + \beta)$$
for every $k$.
It follows from the above two statements that we must have the inequality
$$M(\alpha) \leq M(\alpha + \beta).$$
