# Minimal surfaces with increasing area but bounded Morse index

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  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.

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

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 .