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International audienceWe prove the existence of a countable family of Delaunay type domains \Omega_j in M^n x R, where M^n is the Riemannian manifold S^n or H^n and n is at least 2, bifurcating from the cylinder B^n x R (where B^n is a geodesic ball of radius 1 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. The domains \Omega_j are rotationally symmetric and periodic with respect to the R-axis of the cylinder and as j converges to 0 the domain \Omega_j converges to the cylinder B^n x R

Delaunay type domains for an overdetermined elliptic problem inn× ℝ and ℍn× ℝ

Delaunay type domains for an overdetermined elliptic problem inⁿ× ℝ and ℍⁿ× ℝ