Let $(M,g)$ be a complete and oriented Riemannian 3-manifold without boundary. Given $\Sigma_1$ and $\Sigma_2$ closed surfaces embedded in $M$, where the former is minimal and the latter has CMC $H > 0$, is there a “structure theorem” for the intersection $\Sigma_1 \cap \Sigma_2$? By this I mean a detailed description of the intersection.

If both surfaces are minimal and $p$ is a point of tangency, then Theorem 4.3 in these notes by Chodosh and Mantoulidis give a precise description.

The situation I am interested in is when $\Sigma_2$ is a CMC surface in the round sphere $\mathbb{S}^3$ and $\Sigma_1$ is an equator.