Steklov eigenvalues of nearly hyperspherical domains

We consider Steklov eigenvalues of nearly hyperspherical domains in $\mathbb{R}^{d + 1}$ with $d\ge 3$. In previous work, treating such domains as perturbations of the ball, we proved that the Steklov eigenvalues are analytic functions of the domain perturbation parameter. Here, we compute the first-order term of the asymptotic expansion and show that the first-order perturbations are eigenvalues of a Hermitian matrix, whose entries can be written explicitly in terms of the Pochhammer's and Wigner $3j$-symbols. We analyse the asymptotic expansion and show the following isoperimetric results among domains with fixed volume: (1) for an infinite subset of Steklov eigenvalues, the ball is not optimal, and (2) for a different infinite subset of Steklov eigenvalues, the ball is a stationary point.