Besides the trivial cases of cylinders and spheres, are there any other known examples of non-zero constant mean curvature surfaces which can be represented explicitly in a closed form? I am interested only in 2-dimensional surfaces in Euclidean $\mathbb{R}^3$.
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$\begingroup$ The other surfaces of revolution are the unduloids and noidoids. $\endgroup$– RBega2Commented Oct 23, 2022 at 12:39
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$\begingroup$ gang.umass.edu/gallery/cmc/cmcgallery.html $\endgroup$– Will JagyCommented Oct 23, 2022 at 14:09
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In fact, all CMC tori $f\colon T^2\to\mathbb R^3$ admit closed form expressions in terms of $\theta$-functions on an associated spectral curve $\Sigma.$ The main ideas to show this are as follows: 1) CMC surfaces in euclidean 3-space are characterized by the fact that their Gauss map is harmonic (and non-conformal unless $H=0$, which will be excluded in the following). 2) There is an associated family of CMC surface with the same intrinsic metric, but rotated Hopf differential. This translates to the existence of a family of flat $\mathrm{SL}(2,\mathbb C)$ connections $\nabla^\lambda,\lambda\in\mathbb C^*$ on the torus, which recaptures all informations about $f$. 3) For compact surfaces the family of gauge classes almost contains all informations about $f$, up to so called isospectral deformation which will be discussed below. 4) Since for tori, the fundamental group is abelian the family of flat connections generically splits into to flat line bundle connections, whose gauge classes can be parametrized by meromorphic 1-forms on a hyperelliptic Riemann surface- the associated spectral curve $\Sigma$. 4) The remaining gauge freedom (corresponding to isospectral deformations) is fixed by choosing specifying the eigenline bundle $\mathcal L_p\to\Sigma$ for any point $p\in T^2.$ In fact, it then turns out that $p\to\mathcal L_p$ flows linearly in the affine prim variety of $\Sigma$. 5) The later observation can then be turned into explicit formulas, involving $\theta$ functions on the spectral curve.
Relevant literature is: Pinkall, Sterling: Classification of CMC surfaces, Annals; Hitchin: Harmonic 2-tori in the 3-sphere, JDG and for the explicit formulas: Bobenko: All CMC tori in terms of theta functions, Math. Annalen
