Delaunay type domains for an overdetermined elliptic problem in S^2 x R and ℍ^2 x ℝ

We prove the existence of a countable family of Delaunay type domains Ω_t ⊂ M^n x ℝ, t ∈ ℕ, where M^n is the Riemannian manifold S^n or ℍ^n and n ≥ 2, bifurcating from the cylinder B^n x ℝ (where B^n is a geodesic ball in M^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 {▵g u + λ u = 0 in Ω_t, u = 0 on ∂Ω_t, g(▿u, ν) = const. on ∂Ω_t has a bounded positive solution for some positive constant λ, where g is the standard metric M^n x ℝ. The domains Ω_t are rotationally symmetric and periodic with respect to the ℝ-axis of the cylinder and the sequence {Ω_t}_t converges to the cylinder B^n x ℝ.