The behavior as <$E epsilon~symbol О~0> of the spectrum of the Laplace - Beltrami operator <$E DELTA sup epsilon> is studied on Riemannian manifolds depending on a small parameter <$E epsilon>. They consist of a fixed compact manifold with attached handles whose radii tend to zero as <$E epsilon~symbol О~0>. We consider two cases: when the number of the handles 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 <$E epsilon~symbol О~0>. For these cases we obtain the operators whose spectrum attracts the spectrum of <$E DELTA sup epsilon> as <$E epsilon~symbol О~0>.

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