The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

Details The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem The first eigenvalue of the Laplacian, isoperimetric constants, and the Max Flow Min Cut Theorem

2005-06-13

arxiv.org/abs/math/0506243v2

Mathematics

Differential Geometry

9 pages, added references 6,11,30, to appear in Archiv der Math

Scientific paper

We show how 'test' vector fields may be used to give lower bounds for the Cheeger constant of a Euclidean domain (or Riemannian manifold with boundary), and hence for the lowest eigenvalue of the Dirichlet Laplacian on the domain. Also, we show that a continuous version of the classical Max Flow Min Cut Theorem for networks implies that Cheeger's constant may be obtained precisely from such vector fields. Finally, we apply these ideas to reprove a known lower bound for Cheeger's constant in terms of the inradius of a plane domain.

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