×

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

Citations:

Zbl 0653.53022
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ahlfors, L.V, Lectures on quasiconformal mapping, (1966), Van Nostrand New York · Zbl 0138.06002
[2] Alvarez, O, Theory of strings with boundary, Nuclear phys. B, 216, 125-184, (1983)
[3] Aubin, Th, Meilleurs constantes dans le théorème d’inclusion de Sobolev et un théorème de Fredholm non lineaire pour la transformation conforme de la courboure scalaire, J. funct. anal., 32, 148-174, (1979) · Zbl 0411.46019
[4] Brüning, J, On the compactness of isospectral potentials, Comm. partial differential equations, 9, 687-698, (1984) · Zbl 0547.58039
[5] Gilkey, P.B, Recursion relations and the asymptotic behavior of the eigenvalues of the Laplacian, Compositio math., 38, 201-240, (1979) · Zbl 0405.58050
[6] Gilkey, P, Invariance theory, the heat equation and the Atiyah-Singer index theorem, (1984), Publish or Perish, Inc Wilmington · Zbl 0565.58035
[7] Gordon, C; Wilson, E, Isospectral deformations of compact solvmanifolds, J. differential geom., 19, 241-256, (1984) · Zbl 0523.58043
[8] Guillemin, V; Kazhdan, D, Some inverse spectral results for negatively curved 2-manifolds, Topology, 19, 301-312, (1980) · Zbl 0465.58027
[9] Kac, M, Can one hear the shape of a drum?, Amer. math. monthly, 73, 1-23, (1966) · Zbl 0139.05603
[10] McKean, H.P; Singer, I.M, Curvature and the eigenvalues of the Laplacian, J. differential geom., 1, 43-69, (1967) · Zbl 0198.44301
[11] {\scR. Melrose}, Isospectral drumheads are compact in C∞, preprint.
[12] Mumford, D, A remark on Mahler’s compactness theorem, (), 289-294 · Zbl 0215.23202
[13] Onofri, E, On the positivity of the effective action in a theory of random surfaces, Comm. math. phys., 86, 321-326, (1982) · Zbl 0506.47031
[14] Osgood, B; Phillips, R; Sarnak, P, Extremals of determinants of Laplacians, J. funct. anal., 80, 148-211, (1988) · Zbl 0653.53022
[15] Polyakov, A, Quantum geometry of bosonic strings, Phys. lett. B, 103, 207-210, (1981)
[16] Polyakov, A, Quantum geometry of fermionic strings, Phys. lett. B, 103, 211-213, (1981)
[17] Smith, L, The asymptotics of the heat equation for a boundary value problem, Invent. math., 63, 467-493, (1981) · Zbl 0487.35012
[18] Sunada, T, Riemannian coverings and isospectral manifolds, Ann. of math., 121, 169-186, (1985) · Zbl 0585.58047
[19] Vigneras, M.F, Variétés riemanniennes isospectrales et non isométrique, Ann. of math., 112, 21-32, (1980) · Zbl 0445.53026
[20] {\scS. Wolpert}, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Phys., to appear · Zbl 0629.58029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.