**Question.** What restrictions are known on closed manifolds $(M^3,g)$ that contain a sequence of embedded minimal surfaces $(\Sigma_j \mid j \in \mathbf{N})$ with
\begin{equation}
\mathrm{area} \, \Sigma_j \to \infty \, \, \text{ but } \, \, \mathrm{index} \, \Sigma_j \leq C
\end{equation}
for some constant $C$ and all $j$?

There are no topological restrictions: Colding and Minicozzi [1] show that for any manifold $M$ there is an open set of metrics and a sequence of tori $(T_j)$ with \begin{equation} \mathrm{index} \, T_j = 0 \, \, \text{ but } \, \, \mathrm{area} \, T_j \to \infty. \end{equation} They also remark that such a sequence could not exist if the metric has positive Ricci curvature; however I am a bit hazy about the link between index and genus. Is there a bound for the genus of a minimal surface $\Sigma \subset M^3$ in terms of $\mathrm{index} \, \Sigma$ in this case? If indeed $\mathrm{Ric}_g > 0 $ is forbidden, I'd be interested in any information available beyond this.

[1] Tobias Colding and William Minicozzi. Examples of embedded minimal tori without area bounds. *International Mathematics Research Notices.* No. 20 (1999) pp. 1097-1100.