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

Pieralberto Sicbaldi (1)

1 I2M - Institut de Mathématiques de Marseille

We prove the existence of extremal domains for the first eigenvalue of the Laplace-Beltrami operator in some compact Riemannian manifolds, with volume close to the volume of the manifold. If the first (positive) eigenfunction F 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 F attains its maximum. If F is a constant function and the dimension of the manifold is at least 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.

Domaines: Géométrie différentielle [math.DG]; Mathématiques [math]; Equations aux dérivées partielles [math.AP]

Pieralberto Sicbaldi. Extremal domains of big volume for the first eigenvalue of the Laplace-Beltrami operator in a compact manifold. Annales de l'Institut Henri Poincaré C, Analyse non linéaire, 2014, 31 (6), pp.1231-1265. ⟨hal-00480303v2⟩