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Diffusion on manifolds with small handles. (English) Zbl 0574.58032

The authors show that local topological perturbations of a Riemannian manifold M may only slightly influence the spectrum of the corresponding Laplace-Beltrami operator \(\Delta\). The proof is based on the result obtained in this paper that the Brownian motion on M ”hardly notice” a small tubular neighborhood of any compact submanifold of codimension larger than one. The research continues the authors’ previous study [see Comment. Math. Helv. 56, 83-102 (1981; Zbl 0473.53037); Math. Z. 184, 435-448 (1983; Zbl 0525.53054)].
Reviewer: Y.Kifer

MSC:

58J65 Diffusion processes and stochastic analysis on manifolds
58J35 Heat and other parabolic equation methods for PDEs on manifolds
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