## Compact isospectral sets of surfaces.(English)Zbl 0653.53021

Let (M,g) be a compact Riemann surface with smooth boundary dM. Metrics $$g_ v$$ are said to be isospectral if the corresponding Laplacians (with Dirichlet boundary conditions) have the same spectrum. The group of gauge transformations in question is the group of diffeomorphisms. They show that families of isospectral metrics are compact modulo gauge equivalence. The proof is a lovely blend of local and global analysis. The functional determinant [see the next review] is used to normalize the variation across conformal classes; the asymptotics of the heat equation then come into play to provide the requisite Sobolev estimates. In dimension 3, Chang-Yang have shown that isospectral families within a conformal class are compact modulo gauge equivalence; the primary difference between dimensions 2 and 3 is that in dimension 2 the moduli space of conformal structures is well understood; the functional determinant is a non-local invariant which plays a crucial role in the analysis.
Reviewer: P.Gilkey

### MSC:

 53C20 Global Riemannian geometry, including pinching 58J50 Spectral problems; spectral geometry; scattering theory on manifolds

Zbl 0653.53022
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### References:

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