Recently, Andrews and Li announced a complete classification of CMC ($H=const.$) tori in $S^3$, confirming a conjecture of Pinkall and Sterling. Their main result is that any such torus is rotationally symmetric, and this reduces the problem to a previous classification due to Perdomo, also studied by Hynd, Park and McCuan. [As a side note, some of the techniques used are similar to Brendle's solution of the Lawson Conjecture, that states that the only minimal ($H=0$) torus in $S^3$, up to congruences, is the Clifford torus $S^1(\tfrac{1}{\sqrt2})\times S^1(\tfrac{1}{\sqrt2})$.] It is also known that there are CMC tori in $S^3$ that are not congruent to the family $S^1(r)\times S^1(\sqrt{1-r^2})$ of Clifford tori, so, in a certain sense, their only (continuous) symmetries are encoded in an isometric circle action, as confirmed by Andrews' and Li's result.

Given this context, my question is:

Are there open questions, conjectures, etc., regarding existence or non-existence of CMC hypersurfaces with symmetries in highly symmetric compact manifolds, analogously to the above case of CMC tori in $S^3$? Actually, I'd be particularly interested in questions regarding non-minimal CMC embeddings ($H=const\neq 0$).

To be (slightly) more precise, I am interested in knowing if there are questions of the form "Let $(M,g)$ be a compact manifold with many symmetries (i.e., a "big" Lie group acts isometrically), and consider embeddings of some manifold $N$ into $M$. Then if $N\subset M$ is CMC, it must have certain symmetries (and perhaps cannot have other symmetries)." In other words, are there any analogues of (or questions similar to) the conjecture of Pinkall and Sterling that CMC tori in $S^3$ are rotationally symmetric, but for other (perhaps higher dimensional) compact manifolds? Perhaps some of these questions are disguised as questions about Mean Curvature Flow?

Notice also that if $N\subset M$ is an orbit of an isometric group action, then it is automatically CMC. However, there could be other CMC embeddings of $N$ into $M$, not congruent to any orbits (e.g., with less symmetries, as the tori above). Should one expect any general behavior for this other CMC embeddings? Are there any known results, questions or conjectures in this direction?

I apologize for the somewhat vague question, but none of the above references seems to risk any further conjectures or indicates natural extensions of their results. Also, a quick google search and inverse citations on MathSciNet don't give much in terms of open questions. I was wondering if there is a reason behind it... Any references to recent survey-type papers related would also be very appreciated!

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    $\begingroup$ I have no answer to your question, but an interesting class of highly symmetric manifold that generalizes the sphere are the symmetric spaces ($\mathbb{CP}^2$ could be an interesting case). However, since most symmetric spaces are higher-dimensional, one should first ask what happens for $\mathbb{S}^n$ when $n>3$. $\endgroup$ – Benoît Kloeckner Oct 3 '12 at 7:05
  • $\begingroup$ @Bernoit: Yes, indeed. A similar picture to $S^3$ is true for $S^{n+1}$, namely, there is a 1-parameter family of CMC "Clifford tori" $S^a\times S^b\subset S^{n+1}$, where $a+b=n$, which are orbits of $SO(a+1)\times SO(b+1)$. These are very symmetric CMC hypersurfaces of the sphere, and it can be proven that there are also non-congruent less symmetric CMC embeddings, see arxiv.org/abs/0905.2128 A classification, however, seems out of reach. A similar construction was also carried out for Berger spheres. I agree that looking at other symmetric spaces could be of interest! $\endgroup$ – Renato G. Bettiol Oct 3 '12 at 14:01

From my understanding there are two main directions of research related to your question.

1) There is a lot of interest in seeing how much of the classical theory of CMC and minimal surfaces that is known to hold in the three-dimensional space forms continues to hold in the other Thurston geometries (or more generally -- three dimensional metric lie groups). W. H. Meeks and his collaborators have been pretty active recently in this area. See for instance this survey article by Meeks and Perez -- which contains a section on open problems.

2) The other main area -- is to try and understand the situation in geometries arising mathematical relativity. The starting point here is an observation of Bray (I believe it is in his thesis) that the solution to the (appropriately formulated) isoperimetric problem in the Schwarzschild metric are rotationally invariant spheres. Huisken has recently connected this to important questions in mathematically general relativity and so has attracted a lot of attention recently. I refer you to this preprint of Brendle and also this preprint of Brendle and Eichmair.


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