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

摘要

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 ≥ 2. The volume of such domains is close to the volume of the manifold. If the first eigenfunction ф0 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 Φ0. If ф0 is a constant function and n ≥ 4， these domains are close to the complement of geodesic balls centered at a nondegenerate critical point of the scalar curvature.