  We consider the Dirichlet-to-Neumann operator (DNO) on nearly-hyperspherical
domains in dimension greater than 3. Treating such domains as perturbations of
the ball, we prove the analytic dependence of the DNO on the shape perturbation
parameter for fixed perturbation functions. Consequently, we conclude that the
Steklov eigenvalues are analytic in the shape perturbation parameter as well.
To obtain these results, we use the strategy of Nicholls and Nigam (2004), and
of Viator and Osting (2020); we transform the Laplace-Dirichlet problem on the
perturbed domain to a more complicated, parameter-dependent equation on the
ball, and then geometrically bound the Neumann expansion of the transformed
DNO. These results are a generalization of the work of Viator and Osting (2020)
for dimension 2 and 3.
