# Asymptotics of constant mean curvature surfaces

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  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.

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

First of all, and similar to the case of minimal surfaces, one has to prove that the conformal type of a complete CMC surface of finite type in $$\mathbb R^3$$ is that of a punctured Riemann surface (if I remember correctly this was done by Meeks). The next to show might be that the Hopf differential, i.e. the $$(2,0)$$-part of the second fundamental form, is a meromorphic differential. Then, one can investigate solutions of the Gauss-Codazzi equations on the punctured disc, depending on the order of the pole/zero of the Hopf differential. I guess using the maximum principle one can reduce to the case of second order pole. (For example, if the Hopf differential has no pole at all but the metric degenerates one should be able to prove that the surface is a piece of a sphere...). Finally, one needs to analyse solutions with second order poles. Note that in this situation one has natural coordinates (e.g., provided by $$\mathbb C/2\pi i\mathbb Z$$) for which the Hopf differential is constant, and the Gauss-Codazzi equations become the $$sinh-Gordon$$ equations. Being asymptotically rotational just means that the metric becomes more and more constant along the vertical curves corresponding to the cylinder period. I do not know how to treat the details in this situation without looking into the paper by Korevaar-Kusner-Solomon, but I would guess it also builds on the maximum principle.
I would like to add two additional comments: first of all, the unduloid are elements in a continous family of embedded CMC surfaces which start at a chain of sphere and end at the cylinder. In the latter case, the conformal factor for the metric just becomes constant. Secondly, there has been the famous Lawson conjecture about embedded minimal tori, which has been resolved by Brendle in 2012: All embedded minimal tori in the 3-sphere are rotationally invariant. These are precisely given by the $$\mathbb S^3$$ counterparts of the cylinder and the unduloids.
• I appreciate your interesting comments, thank you for taking the time to write this up. Would you mind clarifying some points? Perhaps you could explain how you expect the maximum principle to imply symmetry? (No need to go into details.) Note that Korevaar--Kusner--Solomon argue in a different way, using reflection principles. Also, could you explain how the third paragraph relates to the rest of your answer? One more point, if I may: what do you mean when you say that the Clifford torus is the '$\mathbf{S}^3$ counterpart of the cylinder and the unduloids'? Apr 15, 2021 at 17:46
• Concerning the third paragraph: the Lawson correspondence gives a correspondence between CMC in $\mathbb R^3$ and CMC and minimal surfaces in $S^3.$ Ususally it destroys closing periods, but in the case of these tori in $S^3$, one period stays closed and you obtain a CMC cylinder in $\R^3.$ Note that the correspondence also holds between some for some non-embedded tori respectively cylinders (nodoids). So there seems to be a relationship between the fact that a cylinder in $\mathbb R^3$ and a torus in $\mathbb S^3$ is embedded. Apr 16, 2021 at 7:15