Abstract
On a compact Riemannian manifold (M, g) the Laplace operator \( \Delta = {g^{ - 1/2}}{\partial _i}\left( {{g^{1/2}}{g^{ij}}{\partial _j}} \right) \)acting on functions have discrete spectrum: \( 0 < {\lambda _1} \le {\lambda _2} \le \ldots \)corresponding to the eigenfunctions
.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
D. R. Adams, A sharp inequality of J. Moser for higher order derivatives, Annals of Math. 128(1988), 385–398.
M. Anderson, Remarks on the compactness of isospectral sets in low dimensions, preprint.
T. Aubin, Equations différentielles non linéaire st problème de Yamabe concemant la courbure scalaire, J. Math. Pures Appli. 55, 269–296.
T. Aubin, Meilleures constantes dans le théorèm d’inclusion de Sobolev et un théorèm de Fred-holm nonlinéaire pour la transformation conforme de la courbure scalaire, J. Funct. Anal. 32(1979), 148–174.
W. Beckner, Sharp Sobolev inequalities on the sphere and the Moser-Ttudinger inequality, preprint.
T. Branson, S.-Y. A. Chang and P. Yang, Estimates and extremals for zeta function determinants in 4-manifolds, preprint.
T. Branson and B. Ørsted, Conformal indices of Riemannian manifolds, Compositio Math. 60, 261–293.
T. Branson and B. Ørsted, Explicit functional determinants in four dimensions, to appear, Proc. Amer. Math. Soc.
R. Brooks, P. Perry, and P. Petersen, Compactness and finiteness theorems for isospectral manifolds, preprint.
R. Brooks, P. Perry, and P. Yang, Isospectral sets of conformally equivalent metrics, Duke Jour. Math. 58, 131–150.
J. Bruning, On the compactness of isospectral potentials, Comm. in Part. Diff. Equations 9, 687–698.
S.-Y. A. Chang and P. Yang, Compactness of isospectral conformal metrics on S 3, Comment. Math. Helvetici 64(1989), 363–374.
S.-Y. A. Chang and P. Yang, Isospectral onformal metrics on 3-manifolds, Jour. Amer. Math. Society 3, 117–145.
S.-Y. A. Chang and P. Yang, Prescribing Gaussian curvature on S 2, Acta Math. 159, 215–259.
S.-Y. A. Chang and P. Yang, Conformal deformation of metrics on S 2, Jour. Differential Geometry 27, 259–296.
CY5] S.-Y. A. Chang and P. Yang, A perturbation result in prescribing scalar curvature on S n, preprint.
P. Gilkey, Leading terms in the asymptotics of the heat equation, Contemporary Math. 73, 79–85.
C. Gordon and E. Wilson, Isospectral deformations of compact solvmanifolds, Jour. Differential Geometry 19, 241–256.
M. Gursky, Cal Tech Thesis, in preparation.
R. Lundelius, Stanford thesis.
H. McKean and I. Singer, Curvature and the eigenvalues of the Laplacian, Jour. Diff. Geometry 1, 43–69.
R. Melrose, Isospectral sets of drumheads are compact in Cx221E, preprint.
E. Onofri, On the positivity of the effective action in a theory of random surfaces, Commun. Math. Phys. 86, 321–326.
B. Osgood, R. Phillips, and P. Sarnak, Extremals of determinants of Laplacians, J. Funct. Anal. 80(1988), 148–211.
B. Osgood, R. Phillips, and P. Sarnak, Compact isospectral sets of surfaces, J. Funct. Anal. 80(1988), 212–234.
B. Osgood, R. Phillips, and P. Sarnak, Moduli space, heights, and isospectral set of plane domains, Annals of Math. 129, 293–362.
T. Parker and S. Rosenberg, Invariants of conformal Laplacians, Jour. Diff. Geometry 25, 199–222.
A. Polyakov, Quantum geometry of Bosonic strings, Phys. Lett. B 103(1981), 207–210.
D. Ray and I. Singer, R-torsion and the Laplacian on Riemannian manifolds, Advances in Math. 7, 145–210.
R. Schoen, Conformal deformation of a Riemannian metric to constant scalar curvature, Jour. Diff. Geometry 21, 479–495.
T. Sunada, Riemannian coverings and isospectral manifolds, Annals of Math. 121, 169–186.
S. Wolpert, Asymptotics of the spectrum and the Selberg zeta function on the space of Riemann surfaces, Comm. Math. Physics 112, 283–315.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1991 Springer-Verlag Tokyo
About this paper
Cite this paper
Chang, SY.A., Yang, P.C. (1991). Spectral Invariants of Conformal Metrics. In: Igari, S. (eds) ICM-90 Satellite Conference Proceedings. Springer, Tokyo. https://doi.org/10.1007/978-4-431-68168-7_5
Download citation
DOI: https://doi.org/10.1007/978-4-431-68168-7_5
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-70084-5
Online ISBN: 978-4-431-68168-7
eBook Packages: Springer Book Archive