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 $\alpha + \beta$?



Behaviour of mass for currents with disjoint supports
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
$$M(\alpha) \leq M(\alpha + \beta).$$


