Identification of extremal sets for the Bergman isoperimetric inequality

Determine whether the minimizers (extremal sets) of the Bergman isoperimetric inequality in B^n ⊂ C^n are precisely the Euclidean balls centered at the origin and their holomorphic images under automorphisms of the unit ball.

Background

In Euclidean and hyperbolic geometries, spheres or geodesic spheres are known minimizers for isoperimetric problems. For the Bergman ball, the authors highlight a specific candidate class of minimizers—Euclidean balls centered at the origin and their holomorphic images—and present evidence supporting this possibility via their stability result for nearly spherical sets.

Confirming the exact form of extremal sets would complete the characterization of the isoperimetric profile in the Bergman ball and align the complex-analytic setting with classical geometric results by identifying the optimal shapes under the Bergman metric.

References

A particularly interesting question is whether the extremal sets for such an isoperimetric inequality are Euclidean balls centered at the origin and their holomorphic images. Evidence supporting this conjecture is provided in this paper.

— Isoperimetric inequality for nearly spherical domains in the Bergman ball in $\mathbb{C}^n$  (2502.00891 - Kalaj, 2 Feb 2025) in Section 1. Introduction (Extensions to hyperbolic space)