Abstract. We study Serrin's overdetermined boundary value problemin subdomains Ω of the round unit sphere S N ⊂ R N +1 , where ∆ S N denotes the Laplace-Beltrami operator on S N . A subdomain Ω of S N is called a Serrin domain if it admits a solution of this overdetermined problem. In our main result, we construct Serrin domains in S N , N ≥ 2 which bifurcate from symmetric straight tubular neighborhoods of the equator. Our result provides the first example of Serrin domains in S N which are not bounded by geodesic spheres.

Let (M, g) be a compact Riemannian manifold of dimension N , N ≥ 2. In this paper, we prove that there exists a family of domains (Ω ε ) ε∈(0,ε0) and functions u ε such thatwhere ν ε is the unit outer normal of ∂Ω ε . The domains Ω ε are smooth perturbations of geodesic balls of radius ε. If, in addition, p 0 is a non-degenerate critical point of the scalar curvature of g then, the family (∂Ω ε ) ε∈(0,ε0) constitutes a smooth foliation of a neighborhood of p 0 . By considering a family of domains Ω ε in which (0.1) is satisfied, we also prove that if this family converges to some point p 0 in a suitable sense as ε → 0, then p 0 is a critical point of the scalar curvature. A Taylor expansion of he energy rigidity for the torsion problem is also given.

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We study hypersurfaces with fractional mean curvature in N-dimensional Euclidean space. These hypersurfaces are critical points of the fractional perimeter under a volume constraint. We use local inversion arguments to prove existence of smooth branches of multiply-periodic hypersurfaces bifurcating from suitable parallel hyperplanes.