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

Abstract

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 ${\varphi }_0$ 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 ${\varphi }_0$. If ${\varphi }_0$ 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.