  Using the local picture of the degeneration of sequences of minimal surfaces
developed by Chodosh, Ketover and Maximo we show that in any closed Riemannian
3-manifold $(M,g)$, the genus of an embedded CMC surface can be bounded only in
terms of its index and area, independently of the value of its mean curvature.
We also show that if $M$ has finite fundamental group, the genus and area of
any non-minimal embedded CMC surface can be bounded in term of its index and a
lower bound for its mean curvature.

Differential Geometry