Index theorems on manifolds with straight ends

Abstract 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.

[1]  D. Grieser,et al.  Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I , 2008, 0809.0925.

[2]  W. Ballmann,et al.  Regularity and index theory for Dirac-Schrödinger systems with Lipschitz coefficients , 2007, math/0703185.

[3]  R. Mazzeo,et al.  The index of Dirac operators on manifolds with fibered boundaries , 2006, math/0609614.

[4]  Albert Y. Zomaya,et al.  Partial Differential Equations , 2007, Explorations in Numerical Analysis.

[5]  I. Belegradek,et al.  Classification of negatively pinched manifolds with amenable fundamental groups , 2004, math/0402268.

[6]  W. Ballmann,et al.  Eigenvalues and holonomy , 2002, math/0207045.

[7]  R. Camporesi,et al.  The continuous spectrum of the Dirac operator on noncompact Riemannian symmetric spaces of rank one , 2001 .

[8]  W. Ballmann,et al.  On the spectral theory of manifolds with cusps , 2001 .

[9]  M. Mitrea.  Generalized dirac operators on nonsmooth manifolds and Maxwell's equations , 2001 .

[10]  G. Carron.  Un théorème de l’indice relatif , 2001 .

[11]  Boris Vaillant.  Index and Spectral Theory for Manifolds with Generalized Fibred Cusps , 2001, math/0102072.

[12]  G. Carron.  Théorèmes de l'indice sur les variétés non-compactes , 2001 .

[13]  Christian Baer.  The Dirac operator on hyperbolic manifolds of finite volume , 2000, math/0010233.

[14]  J. Lott.  Collapsing and Dirac-Type Operators , 2000, math/0005009.

[15]  J. Lott.  On the spectrum of a finite-volume negatively-curved manifold , 1999, math/9908136.

[16]  R. Melrose,et al.  PSEUDODIFFERENTIAL OPERATORS ON MANIFOLDS WITH FIBRED BOUNDARIES , 1998, math/9812120.

[17]  Michael Taylor,et al.  Partial Differential Equations I: Basic Theory , 1996 .

[18]  Steven G. Krantz,et al.  Invariance Theory Heat Equation and Atiyah Singer Index Theorem , 1995 .

[19]  A. W. Knapp,et al.  Cohomological Induction and Unitary Representations (PMS-45) , 1995 .

[20]  P. Gilkey.  On the Index of Geometrical Operators for Riemannian-Manifolds with Boundary , 1993 .

[21]  B. Booss-Bavnbek,et al.  Elliptic Boundary Problems for Dirac Operators , 1993 .

[22]  W. Fulton,et al.  Representation Theory: A First Course , 1991 .

[23]  M. Stern.  L2-index theorems on locally symmetric spaces , 1989 .

[24]  H. Donnelly.  Essential spectrum and heat kernel , 1987 .

[25]  C. Deninger,et al.  Thee-invariant and the spectrum of the Laplacian for compact nilmanifolds covered by Heisenberg groups , 1984 .

[26]  M. Gromov,et al.  Positive scalar curvature and the Dirac operator on complete riemannian manifolds , 1983 .

[27]  H. Moscovici,et al.  L2-index and the Selberg trace formula , 1983 .

[28]  A. Borel,et al.  $L^2$-cohomology of locally symmetric manifolds of finite volume , 1983 .

[29]  S. Zucker.  L2 cohomology of warped products and arithmetic groups , 1982 .

[30]  P. Eberlein.  Lattices in spaces of nonpositive curvature , 1980 .

[31]  M. Gromov.  ALMOST FLAT MANIFOLDS , 1978 .

[32]  H. Karcher,et al.  Jacobi fields and Finsler metrics on compact Lie groups with an application to differentiable pinching problems , 1974 .

[33]  V. K. Patodi,et al.  Spectral asymmetry and Riemannian Geometry. I , 1973, Mathematical Proceedings of the Cambridge Philosophical Society.

[34]  R. Parthasarathy.  Dirac operator and discrete series , 1972 .

[35]  B. Kostant.  Lie Algebra Cohomology and the Generalized Borel-Weil Theorem , 1961 .

[36]  Jiang-Hua Lu,et al.  Progress in Mathematics , 2013 .

[37]  Michael Frazier,et al.  Studies in Advanced Mathematics , 2004 .

[38]  W. Ballmann,et al.  On the Spectral Theory of Surfaces with Cusps , 2003 .

[39]  L’INVARIANT η POUR LES VARI´ET´ES LIPSCHITZIENNES , 2001 .

[40]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[41]  A. W. Knapp,et al.  Cohomological Induction and Unitary Representations , 1995 .

[42]  Werner Ballmann,et al.  Lectures on Spaces of Nonpositive Curvature , 1995 .

[43]  N. Anghel.  An abstract index theorem on non-compact Riemannian manifolds , 1993 .

[44]  Brian H. Bowditch,et al.  Discrete parabolic groups , 1993 .

[45]  D. Bleecker,et al.  The Heat Equation , 1992 .

[46]  M. Stern.  Eta invariants and Hermitian locally symmetric spaces , 1990 .

[47]  W. Müller.  L2-Index and resonances , 1988 .

[48]  Toshikazu Sunada,et al.  Geometry and analysis of manifolds , 1988 .

[49]  W. Müller.  Manifolds with cusps of rank one , 1987 .

[50]  W. Ballmann,et al.  On the differentiability of horocycles and horocycle foliations , 1987 .

[51]  W. Müller.  Manifolds with cusps of rank one : spectral theory and L[2]-index theorem , 1987 .

[52]  Carolyn S. Gordon,et al.  The spectrum of the Laplacian on Riemannian Heisenberg manifolds. , 1986 .

[53]  M. Gromov,et al.  Manifolds of Nonpositive Curvature , 1985 .

[54]  J. Cheeger,et al.  Bounds on the von Neumann dimension of $L\sp 2$-cohomology and the Gauss-Bonnet theorem for open manifolds , 1985 .

[55]  H. Karcher,et al.  Frame flows on manifolds with pinched negative curvature , 1984 .

[56]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[57]  P. Buser,et al.  Gromov's almost flat manifolds , 1981 .

[58]  Aroldo Kaplan,et al.  Fundamental solutions for a class of hypoelliptic PDE , 1980 .

[59]  S. Yau,et al.  Estimates of eigenvalues of a compact Riemannian manifold , 1980 .

[60]  Ernst Heintze,et al.  Geometry of horospheres , 1977 .

[61]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[62]  D. Widder.  The heat equation , 1975 .

[63]  G. Harder.  A Gauss-Bonnet formula for discrete arithmetically defined groups , 1971 .

[64]  S. Agmon.  Lectures on Elliptic Boundary Value Problems , 1965 .