Question

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?

Answer 1

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