General isoperimetric inequality in the Bergman ball

Establish an isoperimetric inequality for arbitrary measurable subsets E of the Bergman ball B^n ⊂ C^n (the unit ball equipped with the Bergman metric), relating the Bergman perimeter P(E) to the Bergman volume μ(E) for all dimensions n ≥ 2, beyond the nearly spherical case treated in this work.

Background

The paper proves a quantitative stability version of the isoperimetric inequality for nearly spherical sets in the Bergman ball, extending Fuglede’s theorem to this complex-analytic setting. However, the full isoperimetric inequality—valid for arbitrary subsets in the Bergman ball—remains unproven.

The Bergman metric on the unit ball in C^n has nonconstant negative curvature and is neither isotropic nor conformally flat, which complicates the classical isoperimetric analysis known from Euclidean and hyperbolic contexts. The authors emphasize that solving the general inequality would determine the isoperimetric profile of the Bergman ball and has connections to problems such as the Lieb–Wehrl entropy conjecture for SU(N,1).