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

Sicbaldi, Pieralberto

Annales de l'I.H.P. Analyse non linéaire, Tome 31 (2014) no. 6, pp. 1231-1265

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 ${\phi }_0$ 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 ${\phi }_0$. If ${\phi }_0$ 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.

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