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.
Best Answer
Positive scalar curvature implies that if $\textrm{index}(\Sigma_j)\leq I$ then $\Sigma_j$ have bounded area and genus. This is proven here https://arxiv.org/pdf/1509.06724.pdf (Theorem 1.3). That paper also contains some other examples related to the Colding--Minicozzi looping example.
A natural generalization is whether or not one can generalize this to a statement like $$ \textrm{area}(\Sigma) + \textrm{genus}(\Sigma) \leq C \, \textrm{index}(\Sigma) $$ where $C$ is some constant independent of $\Sigma$ (universal or $M$ dependent). Some form of this inequality was conjectured by Schoen and Marques--Neves. There has been a lot of work on this problem, but it is still open. See e.g. https://arxiv.org/pdf/1911.09166.pdf, https://mathscinet.ams.org/mathscinet-getitem?mr=2779062, https://mathscinet.ams.org/mathscinet-getitem?mr=3770846 .