We obtain the existence of a smooth 1-parameter family of non-compact domains \(\Omega _s\subset \mathbb {M}^n\times \mathbb {R}\) , \(n\ge 2\) , bifurcating from the straight cylinder \(B_1\times \mathbb {R}\) such that the problem \(\begin{aligned} -\Delta u=1\,\, \text {in}\,\,\Omega _s, \,\, u=0,\,\,\partial _\nu u=\text {constant}\,\,\text {on}\,\,\partial \Omega _s \end{aligned}\) has a bounded solution, where \(\mathbb {M}^n\) is the Riemannian manifold \(\mathbb {S}^n\) or \(\mathbb {H}^n\) , and \(B_1\) is a unit geodesic ball in \(\mathbb {M}^n\) . The domains \(\Omega _s\) are rotationally symmetric and periodic with respect to the \(\mathbb {R}\) -axis of the cylinder. Moreover, we also show that the bifurcation is critical.
