Extremal Domains of Big Volume for the First Eigenvalue of the Laplace-Beltrami Operator in a Compact Manifold

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds of dimension $n \geq 2$, with volume close to the volume of the manifold. If the first (positive) eigenfunction $\phi_0$ of the Laplace-Beltrami operator over the manifold is a nonconstant function, these domains are close to the complement of geodesic balls of small radius whose center is close to the point where $\phi_0$ attains its maximum. If $\phi_0$ is a constant function and $n \geq 4$, these domains are close to the complement of geodesic balls of small radius whose center is close to a nondegenerate critical point of the scalar curvature function.