The Gromov--Ros conjecture in complex hyperbolic space

Let $\CH^m$, $m\ge2$, have holomorphic sectional curvature $-4$.

We prove that its finite-perimeter isoperimetric regions are precisely the geodesic balls, in every complex dimension.

Starting from a volume geometric median, we use exact-volume radial-angular polar stretches.

These maps are globally bi-Lipschitz, and the reduced-boundary area formula expresses their perimeter through an ambient cofactor.

Summing the second derivatives over the coordinate functions of the direction sphere yields the negative of an explicit polynomial in the horizontal and Reeb components of the measure-theoretic normal.

The polynomial is nonnegative for every real dimension $n=2m\ge4$ and vanishes only for a radial normal.

Global minimality therefore forces radiality almost everywhere on the reduced boundary; a BV $U(m)$-invariance argument and a weighted one-dimensional endpoint comparison then force a single ball.

The same trace identity, combined with cutoff, critical-Hessian, and moving-pole arguments, also proves that every smooth bounded fixed-volume stable critical domain is a geodesic ball.

Consequently, the previously conditional convex weighted-Bergman contraction, Faber--Krahn concentration, and Lieb--Wehrl entropy inequalities become unconditional on $\mathbb B^m$ for every $m\ge2$, with their equality cases. {The exact-volume correction admits a basis-free log-partition formulation, whose Hessian trace has an exact Gaussian covariance interpretation.

In the high-weight flat limit, the Bergman contractivity recovers holomorphic Gaussian hypercontractivity.}