Critical domains for the first nonzero Neumann eigenvalue in Riemannian manifolds

The present paper is devoted to geometric optimization problems related to the Neumann eigenvalue problem for the Laplace-Beltrami operator on bounded subdomains $\Omega$ of a Riemannian manifold $(\mathcal{M},g)$. More precisely, we analyze locally extremal domains for the first nontrivial eigenvalue $\mu_2(\Omega)$ with respect to volume preserving domain perturbations, and we show that corresponding notions of criticality arise in the form of overdetermined boundary problems. Our results rely on an extension of Zanger's shape derivative formula which covers the case when $\mu_2(\Omega)$ is not a simple eigenvalue. In the second part of the paper, we focus on product manifolds of the form $\mathcal{M} = \mathbb{R}^k \times \mathcal{N}$, and we classify the subdomains where an associated overdetermined boundary value problem has a solution.