# Isoperimetric inequality in complex hyperbolic space

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
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