On the genus and area of constant mean curvature surfaces with bounded index

TL;DR

This paper proves that in closed 3-manifolds, the genus of embedded constant mean curvature surfaces can be bounded by their index and area, regardless of mean curvature, with additional bounds when the manifold has finite fundamental group.

Contribution

It establishes genus and area bounds for embedded CMC surfaces based on index and area, extending previous degeneration analysis to broader settings.

Findings

Genus of embedded CMC surfaces is bounded by index and area.

In manifolds with finite fundamental group, genus and area depend on index and mean curvature lower bound.

Bounds are independent of the mean curvature value.

Abstract

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.