| Abstract
| - We prove the existence of a countable family of Delaunay type domains \begin{eqnarray} \Omega_t \subset \mathbb{M}^n \times \mathbb{R}, end{eqnarray}t ∈ℕ , where n is the Riemannian manifold n or ℍ n and n ≥ 2 , bifurcating from the cylinder Bn × ℝ (where Bn is a geodesic ball in n) for which the first eigenfunction of the Laplace-Beltrami operator with zero Dirichlet boundary condition also has constant Neumann data at the boundary. In other words, the overdetermined problem \begin{eqnarray} \left\{ \begin{array} {ll} \Delta_g\, u + \gl\, u = 0 &\mbox{in }\; \Omega_t\ u=0 & \mbox{on }\; \partial \Omega_t \ g(bla u, u) = \const. &\mbox{on }\; \partial \Omega_t end{array} \right. end{eqnarray} has a bounded positive solution for some positive constant λ, where g is the standard metric in n × ℝ . The domains Ωt are rotationally symmetric and periodic with respect to the ℝ-axis of the cylinder and the sequence { Ωt } t converges to the cylinder Bn × ℝ .
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