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 ...')
If the homology class $[f(\Sigma)] \in H_2(N)$ of the image of $f: \Sigma \to M$ 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...
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.
