Isoperimetric inequality for nearly spherical domains in the Bergman ball
David Kalaj
TL;DR
The paper tackles a quantitative isoperimetric problem for nearly spherical domains in the Bergman ball $\mathbb{B}_2\subset\mathbb{C}^2$. It expresses boundary deformations as a graph $Z(\omega)=\omega\tanh\left(\frac{r}{2}(1+u(\omega))\right)$ over the unit sphere and proves a Fuglede-type stability: the isoperimetric deficit $D(E)$ controls the $W^{1,2}$-norm of the boundary perturbation $u$, i.e. $D(E) \ge c_1(r_0)\|u\|_{W^{1,2}}^2$ for small $\|u\|_{W^{1,\infty}}$. Central to the argument are (i) a Bergman-perimeter formula $P(E)=\int_{U(z)=c}(1-|z|^2)^{-n-1/2}\sqrt{1-|\langle \nabla U/|\nabla U|, z\rangle|^2}\,d\mathcal{H}(z)$, (ii) a Jacobian analysis for the boundary parameterization, and (iii) a spectral decomposition of the boundary perturbation via spherical harmonics on $\mathbb{S}$, with a specialized estimate for $n=3$ on $\nabla_{i z}$. The results extend stability phenomena known in hyperbolic spaces to the Bergman setting and provide a rigorous quantitative link between geometric deviation and perimeter deficit for Bergman-geometric isoperimetry. The work also clarifies the role of holomorphic barycenters and automorphism invariance in reducing the problem to origin-centered configurations and developing the perimeter expansions needed for the stability estimate.
Abstract
We prove a quantitative isoperimetric inequality for the nearly spherical subset of the Bergman ball in $\mathbb{C}^2$. We prove the Fuglede theorem for such sets. This result is a counterpart of a similar result obtained for the hyperbolic unit ball and it makes the first result on the isoperimetric phenomenon in the Bergman ball.