# Space of embedded minimal surfaces of fixed genus in a generic $3$-manifold

Let $$M^3$$ be a closed, connected and oriented smooth $$3$$-manifold, and fix an integer $$g \geq 1$$. Is it true that for a generic set of Riemannian metrics on $$M$$ the set of closed, connected and orientable embedded minimal surfaces of genus $$g$$ in $$M$$ is compact? If not, is it true for a generic set of metrics of positive scalar curvature?

There is no hope that this holds for every metric, since $$\Sigma_{\gamma} \times \mathbb{S}^1$$ has a sequence of minimal embedded tori winding around $$\mathbb{S}^1$$ as many times as one wishes. Here $$\Sigma_\gamma$$ is a compact Riemannian surface of genus $$\gamma \geq 1$$ endowed with a metric of constant curvature equal to $$0$$, if $$g=1$$, or equal to $$-1$$ if $$g \geq 2$$ (see this paper, for instance).

The set of embedded minimal surfaces $$\Sigma$$ with $$\textrm{genus}(\Sigma) \leq g_0$$ and $$\textrm{area}(\Sigma)\leq A_0$$ is compact for a generic metric (this follows from White's version of the Choi--Schoen compactness theorem https://mathscinet.ams.org/mathscinet-getitem?mr=880951, see also Appendix A: https://mathscinet.ams.org/mathscinet-getitem?mr=1778099).
Edit: The second part of the question (generic PSC metric) is trickier. I think that no counterexample is known but I would guess that one exists. The tricky part is that for $$(M,g)$$ a $$3$$-manifold with a PSC metric, the space of bounded Morse index surfaces is compact. So any counterexample would necessarily have unbounded Morse index (as in the Colding--De Lellis example above). This rules out minimization, which is the tool used to construct the other examples.