The authors consider surfaces \(S\) of constant mean curvature (CMC surfaces) that are surfaces of revolution in \(S^3\). They come in three classes: (1) round spheres, (2) tori with two distinct axis of revolution, the flat CMC tori, (3) tori with only one axis of revolution, the non-flat CMC tori. The author obtains some information on the index of such surfaces. Here the index of \(S\), Ind\((S)\), is the number of negative eigenvalues of the Jacobi operator \(\mathcal I\) acting on \(C^\infty\) functions from \(S\) to \(\mathbb R\). This is the strong index. In constrast to it, the weak index is obtained when we consider \(\mathcal I\) acting on real functions whose integral on \(S\) vanishes. It is well known that they differ at most by one. The nullity of \(S\) is the multiplicity of the zero eigenvalue of \(\mathcal I\).
Before describing the main results of this paper, we still need a definition. For a non-flat torus of revolution, taking a geodesic hemisphere whose boundary is the axis of revolution, the hemisphere interset \(S\) along a curve with an equal finite number of points of maximal and minimal distance from the axis, which are called bulges and necks, respectively.
The authors prove the following.
Theorem. Let \(S\) be a closed CMC \(H\) surface of revolution in \(S^3\). Then either:

1)

\(S\) is a round sphere with index 1 and nullity 3, or

2)

\(S\) is a flat torus with index \(3+2b\) and nullity 4 or 6, or (here \(b\) is the greatest integer less than \(\sqrt{1+e^{2\text{arc sinh}|H|}}\)

3)

\(S\) is a non-flat torus with \(k\) bulges and \(k\) necks and has index at least max\((5,2k+1)\) and nullity at least 5.

A key ingredient fro the proof of part (3) is an application of Courant’s nodal theorem.