New examples of extremal domains for the first eigenvalue of the   Laplace-Beltrami operator in a Riemannian manifold with boundary
  We build new examples of extremal domains with small prescribed volume for the first eigenvalue of the Laplace-Beltrami operator in some Riemannian manifold with boundary. These domains are close to half balls of small radius centered at a nondegenerate critical point of the mean curvature function of the boundary of the manifold, and their boundary intersects the boundary of the manifold orthogonally. 
Jimmy Lamboley, Pieralberto Sicbaldi
math.DG math.AP