Extremal domains of big volume for the first eigenvalue of the Laplace–Beltrami operator in a compact manifold

We prove the existence of new extremal domains for the first eigenvalue of the Laplace–Beltrami operator in some compact Riemannian manifolds of dimension n⩾2n⩾2. The volume of such domains is close to the volume of the manifold. If the first eigenfunction ϕ0ϕ0 of the Laplace–Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of ϕ0ϕ0. If ϕ0ϕ0 is a constant function and n⩾4n⩾4, these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of the scalar curvature.