On the Genus and Area of Constant Mean Curvature Surfaces with Bounded Index

Using the local picture of the degeneration of sequences of minimal surfaces developed by Chodosh et al. (Invent Math 209(3):617–664, [ 1 ]) 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.