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

Analyticity of Steklov Eigenvalues in nearly-hyperspherical domains in \mathbb{R}^{d+1}