A Smooth 1-Parameter Family of Delaunay-Type Domains for an Overdetermined Elliptic Problem in S^n x R and H^n x R

We prove the existence of a non-compact smooth one-parameter family of domains Ωs⊂Mn×R, where Mn denotes the Riemannian manifold Sn or Hn (for n≥2), bifurcating from the straight cylinder B1×R (where B1 is a geodesic unit ball in Mn) such that there exists a positive solution u to (Formula presented.) for some positive constant λ, where g is the standard metric in Mn×R, and ν represents the unit normal vector about ∂Ωs. The domains Ωs are not straight cylinders but are periodic in the direction of R. This improves a previous result by the second and third author.