In 1979, Rick Schoen and Shing-Tung Yau published an Annals paper in which they proved the existence of so-called 'incompressible' minimal surfaces in compact Riemannian manifolds.

Question. Under what conditions is such an incompressible minimal surface $\Sigma^2 \to N^3$ known to be embedded?

The first thing that would come to my mind would be a topological condition, or something with the curvature of $N$, but I'd be interested in abstract answers too. (E.g. 'If you want to prove that such a $\Sigma \subset N$ is embedded, the first thing to try is ...')

Background. I'll give some more background information, which those familiar with the paper need not read.

My foremost interest lies with the three-dimensional case; the setting was the following. Let $N^3$ be a closed, orientable Riemannian manifold. If $\pi_1(N)$ is known to contain a subgroup $G$ abstractly isomorphic to the fundamental group of a Riemann surface $\Sigma_g$ with genus $g \geq 1$, then there exists a minimal immersion $f: \Sigma_g \to N$ with $f_\# (\pi_1(\Sigma_g)) = G$. (The results are actually more general, but I'll restrict to this simpler setting.) Moreover, $f(\Sigma_g)$ has the least area among all maps homotopic to it.

The map $f$ is constructed in two steps, first as a harmonic map by minimizing the Dirichlet energy, which is then made conformal—and thus a branched minimal immersion—by varying the conformal structure. The construction works in arbitrary dimensions, but when $\operatorname{dim} N = 3$ there are no branch points.

If the homology class $[f(\Sigma_g)] \in H_2(N)$ were non-zero, then one could resolve the immersed singularities and minimize area among all surfaces in the same class. But then $H_2(N) \neq 0$, which—it seems to me—kind of defeats the purpose of following the Schoen–Yau approach in the first place...