Stability theorems for the continuous spectrum of a negatively curved manifold

Donnelly, Harold

doi:10.1090/S0002-9947-1981-0603773-0

Stability theorems for the continuous spectrum of a negatively curved manifold
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Trans. Amer. Math. Soc. 264 (1981), 431-448 Request permission

Abstract:

The spectrum of the Laplacian $\Delta$ for a simply connected complete negatively curved Riemannian manifold is studied. The Laplacian ${\Delta _0}$ of a simply connected constant curvature space ${M_0}$ is known up to unitary equivalence. Decay conditions are given, on the metric $g$ and curvature $K$ of $M$, which imply that the continuous part of ${\Delta _0}$ is unitarily equivalent to ${\Delta _0}$.

References

Marcel Berger, Paul Gauduchon, and Edmond Mazet, Le spectre d’une variété riemannienne, Lecture Notes in Mathematics, Vol. 194, Springer-Verlag, Berlin-New York, 1971 (French). MR 0282313
M. Š. Birman, Existence conditions for wave operators, Izv. Akad. Nauk SSSR Ser. Mat. 27 (1963), 883–906 (Russian). MR 0161150
Richard L. Bishop and Richard J. Crittenden, Geometry of manifolds, Pure and Applied Mathematics, Vol. XV, Academic Press, New York-London, 1964. MR 0169148
Jeff Cheeger and Shing Tung Yau, A lower bound for the heat kernel, Comm. Pure Appl. Math. 34 (1981), no. 4, 465–480. MR 615626, DOI 10.1002/cpa.3160340404
Harold Donnelly, Asymptotic expansions for the compact quotients of properly discontinuous group actions, Illinois J. Math. 23 (1979), no. 3, 485–496. MR 537804
Harold Donnelly, Eigenvalues embedded in the continuum for negatively curved manifolds, Michigan Math. J. 28 (1981), no. 1, 53–62. MR 600414
Harold Donnelly, Spectral geometry for certain noncompact Riemannian manifolds, Math. Z. 169 (1979), no. 1, 63–76. MR 546993, DOI 10.1007/BF01214913
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II: Spectral theory. Self adjoint operators in Hilbert space, Interscience Publishers John Wiley & Sons, New York-London, 1963. With the assistance of William G. Bade and Robert G. Bartle. MR 0188745
Philip Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney, 1964. MR 0171038
Sigurđur Helgason, Differential geometry and symmetric spaces, Pure and Applied Mathematics, Vol. XII, Academic Press, New York-London, 1962. MR 0145455

Harmonic analysis on homogeneous spaces

Lars Hörmander, Fourier integral operators. I, Acta Math. 127 (1971), no. 1-2, 79–183. MR 388463, DOI 10.1007/BF02392052
Arne Jensen and Tosio Kato, Asymptotic behavior of the scattering phase for exterior domains, Comm. Partial Differential Equations 3 (1978), no. 12, 1165–1195. MR 512084, DOI 10.1080/03605307808820089
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
V. K. Patodi, Curvature and the eigenforms of the Laplace operator, J. Differential Geometry 5 (1971), 233–249. MR 292114
Mark A. Pinsky, The spectrum of the Laplacian on a manifold of negative curvature. I, J. Differential Geometry 13 (1978), no. 1, 87–91. MR 520603
Louis Robin, Fonctions sphériques de Legendre et fonctions sphéroïdales. Tome III, Collection Technique et Scientifique du C.N.E.T, Gauthier-Villars, Paris, 1959 (French). MR 0109896
Shing Tung Yau, Isoperimetric constants and the first eigenvalue of a compact Riemannian manifold, Ann. Sci. École Norm. Sup. (4) 8 (1975), no. 4, 487–507. MR 397619