Here are questions on constant mean curvature (CMC) spheres which
arose from the study of the isoperimetric problem in three-dimensional, simply connected, non-compact homogeneous manifolds $X$ diffeomorphic to $\mathbb{R}^3$.

It is well known that CMC spheres in $X$ exist exactly for values of the mean curvature $H$ lying in an interval of the form $(H(X),\infty )$ where $H(X)$ is called the critical mean curvature of X. Furthermore, spheres of CMC $H$ are unique up to congruencies, when they exist. This uniqueness leads to a well-defined notion of center of symmetry for CMC spheres. Finally, it is also known that if we take any point $x\in X$, the family of CMC spheres with center $x$ is a real analytic 1-parameter manifold. It is not known (but expected) if CMC spheres are embedded in general (they are known to be Alexandrov embedded).

Is true that CMC spheres in $X$ are embedded?

More specifically, do CMC spheres centered at a point $x\in X$ form a foliation of $X − \{x\}$? (in particular, such CMC spheres
would be embedded, which implies that every CMC sphere in $X$ is embedded).

ArticleIs an originMeeks, W.H., Mira, P., Pérez, J. et al. Constant mean curvature spheres in homogeneous three-manifolds. Invent. math. 224, 147–244 (2021). https://doi.org/10.1007/s00222-020-01008-y

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