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Let $\mathbb{H}_\mathbb{C}^n$ be n-dimensional complex hyperbolic space. This space is a complex analog of hyperbolic space. It is isometric to the quotient of hyperboloid $$|z_0|^2-|z_1|^2-\dots-|z_n|^2=1$$ in $\mathbb{C}^{n+1}$ by $S^1$.

Question 1. Is it known that round balls in $\mathbb{H}_\mathbb{C}^n$ minimize the surface area among all bodies of given volume?

(I am almost sure that the answer is not known.)

Question 2. Was it conjectured somewhere?

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Googling, I found it conjectured here: www.ugr.es/~aros/isoper.pdf –  Ian Agol Sep 21 '12 at 20:50
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The case of CMC surfaces in $\mathbb{CH}^2$ was claimed ams.org/mathscinet-getitem?mr=1123452, but an error was found: ams.org/mathscinet-getitem?mr=1308011 –  Ian Agol Jul 22 '13 at 1:01
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1 Answer

I am pretty sure it is a somewhat reputed conjecture, but I do not have a clear reference where it is stated.

Edit: It is stated in Gromov's "Metric structures for Riemannian and non-Riemannian spaces", 6.28 (1/2+) (in fact, it is stated for all rank one symmetric spaces).

It might be evoked in a paper of Hsiang and Hsiang in Inventiones, where they prove that the isoperimetric domains in products of hyperbolic and euclidean spaces are invariant under the group of all isometries fixing the center of gravity. It seems a reasonable conjecture that this is true in all symmetric spaces of non-positive curvature. That conjecture might be stated in the Hsiang and Hsiang paper, and is a broad generalization of the conjecture you are interested in.

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I heard it from Marcel Berger in 1994 or 1995 when he was lecturing in Stony Brook. I'm pretty sure I saw the problem (also for rank one symmetric spaces) in one of his books. –  alvarezpaiva Jul 21 '13 at 18:43
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