New examples of extremal domains for the first eigenvalue of
the Laplace-Beltrami operator

Pieralberto Sicbaldi (Marseille)

Abstract: In this talk we show the existence of new extremal domains
for the first eigenvalue of the Laplace-Beltrami operator in some compact
Riemannian manifolds. These domains are close to the complement of geodesic
balls of small radius centered in a convenient point of the manifold, which
can be the point where the first positive eigenfunction of the Laplace-Beltrami
operator over the manifold attains its maximum or a nondegenerate critical
point of the scalar curvature function. In particular, the volume of such
extremal domains is big and their topology can be arbitrary. Moreover, we
will see that one can use such examples of extremal domains to construct
other non-trivial examples.