Daniel Grieser (Oldenburg)

Eigenvalue Estimates, Isoperimetric Inequalities and Flows in Networks

Abstract:
The Laplace operator is a basic object in analysis, differential geometry,
probability theory, quantum mechanics, as well as graph theory (in its
discrete form). A central problem is the determination, or at least
estimation, of its lowest eigenvalue in terms of the geometry of
the space on which the Laplacian is defined. One such estimate is
Cheeger's inequality, which relates the lowest eigenvalue to a
quantity h which closely resembles the classical isoperimetric constant. In
this talk I will give a new perspective on this quantity h by relating it to
the problem of flowing a maximum amount of fluid through the space given
certain local capacity constraints. This relation involves a continuous
analogue of the classical Max Flow Min Cut Theorem
in network theory. From this new perspective we get effective ways of
estimating h and new proofs of Cheeger's inequality as well as
inequalities of the lowest eigenvalue in terms of the inradius.