On spectrum of Riemannian manifolds with attached thin handles

Abstract. The behavior as ε → 0 of the spectrum of the Laplace-Beltrami operator ∆ε is studied on Riemannian manifolds M ε depending on a small pa-rameter ε. They consist of a fixed compact manifold with attached handles whose radii tend to zero as ε → 0. We consider two cases: when the number of the han-dles is fixed and their lengthes are also fixed and when the number of the handles tend to infinity and their lengthes tend to zero as ε → 0. For these cases we obtain the operators whose spectrum attracts the spectrum of ∆ε as ε → 0