We briefly describe the variational characterization of profile curves of invariant constant mean curvature surfaces immersed in a 3-dimensional semi-Riemannian space form \(M_r^3(\rho)\) which was introduced in (Arroyo, Garay & ---, 2018).

It turns out that these surfaces are binormal evolution surfaces spanned by a critical curve of an extension of a Blaschke's variational problem.

Theorem. (Arroyo, Garay & ---, 2018) Let \(S\) be an invariant surface with CMC \(H\). Then, locally, the profile curve of \(S\) is critical for
$$\mathbf{\Theta}_\mu(\gamma)=\int_\gamma \sqrt{\kappa-\mu}\,ds$$ with \(\lvert\mu\rvert=\lvert H\rvert\). Conversely, \(S\) can be locally obtained by the binormal flow of a critical curve.

Although above characterization is local in nature, it can be used to obtain global properties of CMC invariant surfaces. In particular, restricting ourselves to critical curves in \(\mathbb{S}^2(\rho)\), we obtain the following result from (Arroyo, Garay & ---, 2019).

Theorem. (Arroyo, Garay & ---, 2019) There exist non-trivial closed critical curves in \(\mathbb{S}^2(\rho)\), for any value of \(\mu\). Moreover, if the curve is also embedded, then \(\mu\neq -\sqrt{\rho/3}\) is negative.

The translation of this theorem to Delaunay surfaces in \(\mathbb{S}^3(\rho)\) coincides with previous results of Ripoll and Perdomo. Moreover, it allows us to verify the Lawson's conjecture (proved by Brendle).