Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-25T08:00:10.409Z Has data issue: false hasContentIssue false

Index theorems on manifolds with straight ends

Published online by Cambridge University Press:  15 October 2012

Werner Ballmann
Affiliation:
Mathematisches Institut, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany (email: [email protected]) Max-Planck-Institut für Mathematik, Vivatsgasse 7, 53111 Bonn, Deutschland, Germany
Jochen Brüning
Affiliation:
Institut für Mathematik, Humboldt–Universität, Rudower Chaussee 5, 12489 Berlin, Germany (email: [email protected])
Gilles Carron
Affiliation:
Département de Mathématiques, Université de Nantes, 2 rue de la Houssiniére, BP 92208, 44322 Nantes Cedex 03, France (email: [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study Fredholm properties and index formulas for Dirac operators over complete Riemannian manifolds with straight ends. An important class of examples of such manifolds are complete Riemannian manifolds with pinched negative sectional curvature and finite volume.

Type
Research Article
Copyright
Copyright © Foundation Compositio Mathematica 2012

References

[Agm65]Agmon, S., Lectures on elliptic boundary value problems (Van Nostrand, Princeton, NJ, 1965).Google Scholar
[Ang93]Anghel, N., An abstract index theorem on non-compact Riemannian manifolds, Houston J. Math. 19 (1993), 223237.Google Scholar
[APS75]Atiyah, M. F., Patodi, V. K. and Singer, I. M., Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 4369.CrossRefGoogle Scholar
[APS76]Atiyah, M. F., Patodi, V. K. and Singer, I. M., Spectral asymmetry and Riemannian geometry. III, Math. Proc. Cambridge Philos. Soc. 79 (1976), 7199.CrossRefGoogle Scholar
[Bar00]Bär, C., The Dirac operator on hyperbolic manifolds of finite volume, J. Differential Geom. 54 (2000), 439488.CrossRefGoogle Scholar
[Bal95]Ballmann, W., Lectures on spaces of nonpositive curvature (Birkhäuser, Basel, 1995) (with an appendix by Misha Brin; DMV Seminar, 25).CrossRefGoogle Scholar
[BBB87]Ballmann, W., Brin, M. and Burns, K., On the differentiability of horocycles and horocycle foliations, J. Differential Geom. 26 (1987), 337347.CrossRefGoogle Scholar
[BB01]Ballmann, W. and Brüning, J., On the spectral theory of manifolds with cusps, J. Math. Pures Appl. (9) 80 (2001), 593625.CrossRefGoogle Scholar
[BB03]Ballmann, W. and Brüning, J., On the spectral theory of surfaces with cusps, in Geometric analysis and partial differential equations (Springer, Berlin, 2003), 1337.Google Scholar
[BBC03]Ballmann, W., Brüning, J. and Carron, G., Eigenvalues and holonomy, Int. Math. Res. Not. IMRN (2003), 657665.CrossRefGoogle Scholar
[BBC08]Ballmann, W., Brüning, J. and Carron, G., Regularity and index theory for Dirac–Schrödinger systems with Lipschitz coefficients, J. Math. Pures Appl. (9) 89 (2008), 429476.CrossRefGoogle Scholar
[BGS85]Ballmann, W., Gromov, M. and Schroeder, V., Manifolds of nonpositive curvature, Progress in Mathematics, vol. 61 (Birkhäuser Boston, Boston, MA, 1985).CrossRefGoogle Scholar
[BM83]Barbasch, D. and Moscovici, H., L 2-index and the Selberg trace formula, J. Funct. Anal. 53 (1983), 151201.CrossRefGoogle Scholar
[BK06]Belegradek, I. and Kapovitch, V., Classification of negatively pinched manifolds with amenable fundamental groups, Acta Math. 196 (2006), 229260.CrossRefGoogle Scholar
[BW93]Booß-Bavnbek, B. and Wojciechowski, K., Elliptic boundary problems for Dirac operators (Birkhäuser, Basel, 1993).CrossRefGoogle Scholar
[Bow93]Bowditch, B. H., Discrete parabolic groups, J. Differential Geom. 38 (1993), 559583.CrossRefGoogle Scholar
[BK84]Brin, M. and Karcher, H., Frame flows on manifolds with pinched negative curvature, Compositio Math. 52 (1984), 275297.Google Scholar
[BK81]Buser, P. and Karcher, H., Gromov’s almost flat manifolds, Astérisque 81 (1981).Google Scholar
[CP02]Camporesi, R. and Pedon, E., The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one, Proc. Amer. Math. Soc. 130 (2002), 507516.CrossRefGoogle Scholar
[Car01a]Carron, G., Un théorème de l’indice relatif, Pacific J. Math. 198 (2001), 81107.CrossRefGoogle Scholar
[Car01b]Carron, G., Théorèmes de l’indice sur les variétés non-compactes, J. Reine Angew. Math. 541 (2001), 81115.Google Scholar
[CG85]Cheeger, J. and Gromov, M., Bounds on the von Neumann dimension of L 2-cohomology and the Gauss–Bonnet theorem for open manifolds, J. Differential Geom. 21 (1985), 134.CrossRefGoogle Scholar
[DS84]Deninger, C. and Singhof, W., The e-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups, Invent. Math. 78 (1984), 101112.CrossRefGoogle Scholar
[Don87]Donnelly, H., Essential spectrum and heat kernel, J. Funct. Anal. 75 (1987), 362381.CrossRefGoogle Scholar
[Ebe80]Eberlein, P., Lattices in spaces of nonpositive curvature, Ann. of Math. (2) 111 (1980), 435476.CrossRefGoogle Scholar
[FH91]Fulton, W. and Harris, J., Readings in mathematics, in Representation theory. A first course, Graduate Texts in Mathematics, vol. 129 (Springer, New York, NY, 1991).Google Scholar
[GT83]Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order, second edition, Grundlehren der Mathematischen Wissenschaften, vol. 224 (Springer, Berlin, 1983).Google Scholar
[Gil93]Gilkey, P. B., On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math. 102 (1993), 129183.CrossRefGoogle Scholar
[Gil95]Gilkey, P. B., Invariance theory, the heat equation, and the Atiyah–Singer index theorem, Studies in Advanced Mathematics, second edition (CRC Press, Boca Raton, FL, 1995).Google Scholar
[GW86]Gordon, C. S. and Wilson, E. N., The spectrum of the Laplacian on Riemannian Heisenberg manifolds, Michigan Math. J. 33 (1986), 253271.CrossRefGoogle Scholar
[GH09]Grieser, D. and Hunsicker, E., Pseudodifferential operator calculus for generalized ℚ-rank 1 locally symmetric spaces I, J. Funct. Anal. 257 (2009), 37483801.CrossRefGoogle Scholar
[Gro78]Gromov, M., Almost flat manifolds, J. Differential Geom. 13 (1978), 231241.CrossRefGoogle Scholar
[GL83]Gromov, M. and Lawson, H. B. Jr., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Publ. Math. Inst. Hautes Études Sci. 58 (1983), 83196.CrossRefGoogle Scholar
[GKR74]Grove, K., Karcher, H. and Ruh, E. A., Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems, Math. Ann. 211 (1974), 721.CrossRefGoogle Scholar
[Har71]Harder, G., A Gauss–Bonnet formula for discrete arithmetically defined groups, Ann. Sci. Éc. Norm. Supér. (4) 4 (1971), 409455.CrossRefGoogle Scholar
[HI77]Heintze, E. and Im Hof, H. C., Geometry of horospheres, J. Differential Geom. 12 (1977), 481491.CrossRefGoogle Scholar
[Hil00]Hilsum, M., L’invariant η pour les variétés lipschitziennes, J. Differential Geom. 55 (2000), 141.CrossRefGoogle Scholar
[Kap80]Kaplan, A., Fundamental solutions for a class of hypoelliptic PDE generated by composition of quadratic forms, Trans. Amer. Math. Soc. 258 (1980), 147153.CrossRefGoogle Scholar
[KV95]Knapp, A. and Vogan, D., Cohomological induction and unitary representations, Princeton Mathematical Series, vol. 45 (Princeton University Press, Princeton, NJ, 1995).CrossRefGoogle Scholar
[Kos61]Kostant, B., Lie algebra cohomology and the generalized Borel–Weil theorem, Ann. of Math. (2) 74 (1961), 329387.CrossRefGoogle Scholar
[LM89]Lawson, H. B. Jr. and Michelsohn, M.-L., Spin geometry, Princeton Mathematical Series, vol. 38 (Princeton University Press, Princeton, NJ, 1989).Google Scholar
[LMP06]Leichtnam, E., Mazzeo, R. and Piazza, P., The index of Dirac operators on manifolds with fibered boundaries, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 845855.Google Scholar
[LY80]Li, P. and Yau, S. T., Eigenvalues of a compact Riemannian manifold, in Geometry of the Laplace operator, Honolulu, HI, 1979, Proceedings of Symposia in Pure Mathematics, vol. 36 (American Mathematical Society, Providence, RI, 1980), 205239.Google Scholar
[Lot01]Lott, J., On the spectrum of a finite-volume negatively-curved manifold, Amer. J. Math. 123 (2001), 185205.CrossRefGoogle Scholar
[Lot02]Lott, J., Collapsing and Dirac-type operators, Geom. Dedicata 91 (2002), 175196.CrossRefGoogle Scholar
[MM98]Mazzeo, R. and Melrose, R. B., Pseudodifferential operators on manifolds with fibered boundaries, Asian J. Math. 2 (1998), 833866.CrossRefGoogle Scholar
[Mit01]Mitrea, M., Generalized Dirac operators on nonsmooth manifolds and Maxwell’s equations, J. Fourier Anal. Appl. 7 (2001), 207256.CrossRefGoogle Scholar
[Mul87a]Müller, W., L 2-index and resonances, in Geometry and Analysis on manifolds, Lectures Notes in Mathematics, vol. 1339 (Katata, Kyoto, 1987), 203211.Google Scholar
[Mul87b]Müller, W., Manifolds with cusps of rank one, spectral theory and L 2-index theorem, Lecture Notes in Mathematics, vol. 1244 (Springer, Berlin, 1987).Google Scholar
[Par72]Parthasarathy, R., Dirac operator and the discrete series, Ann. of Math. (2) 96 (1972), 130.CrossRefGoogle Scholar
[Ruh82]Ruh, E., Almost flat manifolds, J. Differential Geom. 17 (1982), 114.CrossRefGoogle Scholar
[Ste89]Stern, M., L 2-index theorems on locally symmetric spaces, Invent. Math. 96 (1989), 231282.CrossRefGoogle Scholar
[Ste90]Stern, M., Eta invariants and Hermitian locally symmetric spaces, J. Differential Geom. 31 (1990), 771789.CrossRefGoogle Scholar
[Tay96]Taylor, M., Partial differential equations I. Basic theory, Applied Mathematical Sciences, vol. 115 (Springer, Berlin, 1996).Google Scholar
[Vai01]Vaillant, B., Index- and spectral theory for manifolds with generalized fibred cusps, Bonner Mathematische Schriften, vol. 344 (Universität Bonn, Mathematisches Institut, Bonn, 2001).Google Scholar
[Wol73]Wolf, J. A., Essential self-adjointness for the Dirac operator and its square, Indiana Univ. Math. J. 22 (1973), 611640.CrossRefGoogle Scholar
[Wu88]Wu, H., The Bochner technique in differential geometry, Mathematical Reports, vol. 3, part 2 (Harwood Academic, Chur, Switzerland, 1988), i–xii and 289–538.Google Scholar
[Zuc82]Zucker, S., L 2 cohomology of warped products and arithmetic groups, Invent. Math. 70 (1982), 169218.CrossRefGoogle Scholar