Let $\Sigma^n \subset \mathbf{R}^{n+1}$ be a complete, properly embedded hypersurface with constant, non-zero mean curvature $H \neq 0.$

In the case where the dimension is $n = 2$, $\Sigma$ is non-compact and has finite topology, it was proved by Korevaar, Kusner and Solomon [1] that every end of $\Sigma$ is (exponentially) asymptotic to a cylinder or an unduloid: what they have in common is that both have a central axis of rotational symmetry.

**Question.** What are the 'asymptotic models' for CMC hypersurfaces in higher dimensions $n \geq 3$?

In addition to the higher-dimensional versions of unduloids, there ought to be at least the cylinders of the form $\mathbf{R}^k \times \mathbf{S}^{n+1-k}(\rho)$ with $\rho > 0$, along with other equivariant examples. Do they also support 'unduloid-like' CMC hypersurfaces?

The---admittedly tenuous---analogy with minimal hypersurfaces suggests the possibility of a monotone quantity that would restrict the behaviour at infinity. And indeed the function $\rho \in (0,\infty) \mapsto e^{\lVert H \rVert \rho} \mathcal{H}^n(\Sigma \cap B_\rho)/\omega_n \rho^n$ is increasing, but the exponential factor makes this ill-suited for a large-scale analysis.

[1] Korevaar, Kusner, Solomon. The structure of complete embedded surfaces with constant mean curvature. *J. Differential Geometry*. 30 (1989) 465-503.