Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 19 (2023), 023, 32 pages      arXiv:2110.00390      https://doi.org/10.3842/SIGMA.2023.023

Spectral Asymmetry and Index Theory on Manifolds with Generalised Hyperbolic Cusps

Peter Hochs a and Hang Wang b
a) Institute for Mathematics, Astrophysics and Particle Physics, Radboud University, PO Box 9010, 6500 GL Nijmegen, The Netherlands
b) School of Mathematical Sciences, East China Normal University, No. 500, Dong Chuan Road, Shanghai 200241, P.R. China

Received June 22, 2022, in final form March 28, 2023; Published online April 20, 2023

Abstract
We consider a complete Riemannian manifold, which consists of a compact interior and one or more $\varphi$-cusps: infinitely long ends of a type that includes cylindrical ends and hyperbolic cusps. Here $\varphi$ is a function of the radial coordinate that describes the shape of such an end. Given an action by a compact Lie group on such a manifold, we obtain an equivariant index theorem for Dirac operators, under conditions on $\varphi$. These conditions hold in the cases of cylindrical ends and hyperbolic cusps. In the case of cylindrical ends, the cusp contribution equals the delocalised $\eta$-invariant, and the index theorem reduces to Donnelly's equivariant index theory on compact manifolds with boundary. In general, we find that the cusp contribution is zero if the spectrum of the relevant Dirac operator on a hypersurface is symmetric around zero.

Key words: equivariant index; Dirac operator; noncompact manifold; cusp.

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References

  1. Anghel N., Remark on Callias' index theorem, Rep. Math. Phys. 28 (1989), 1-6.
  2. Anghel N., An abstract index theorem on noncompact Riemannian manifolds, Houston J. Math. 19 (1993), 223-237.
  3. Anghel N., On the index of Callias-type operators, Geom. Funct. Anal. 3 (1993), 431-438.
  4. Anghel N., Fredholmness vs. spectral discreteness for first-order differential operators, Proc. Amer. Math. Soc. 144 (2016), 693-701.
  5. Atiyah M.F., Elliptic operators and compact groups, Lecture Notes in Math., Vol. 401, Springer, Berlin, 1974.
  6. Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69.
  7. Atiyah M.F., Singer I.M., The index of elliptic operators. III, Ann. of Math. 87 (1968), 546-604.
  8. Baier P.D., An index theorem for open manifolds whose ends are warped products, Abh. Math. Sem. Univ. Hamburg 72 (2002), 269-282.
  9. Ballmann W., Brüning J., On the spectral theory of surfaces with cusps, in Geometric Analysis and Nonlinear Partial Differential Equations, Springer, Berlin, 2003, 13-37.
  10. Ballmann W., Brüning J., Carron G., Index theorems on manifolds with straight ends, Compos. Math. 148 (2012), 1897-1968, arXiv:1011.2290.
  11. Bär C., The Dirac operator on hyperbolic manifolds of finite volume, J. Differential Geom. 54 (2000), 439-488, arXiv:math.DG/0010233.
  12. Bär C., Ballmann W., Boundary value problems for elliptic differential operators of first order, Surv. Differ. Geom., Vol. 1, Int. Press, Boston, MA, 2012, 1-78, arXiv:1101.1196.
  13. Barbasch D., Moscovici H., $L^{2}$-index and the Selberg trace formula, J. Funct. Anal. 53 (1983), 151-201.
  14. Baum P., Connes A., Higson N., Classifying space for proper actions and $K$-theory of group $C^\ast$-algebras, in $C^\ast$-Algebras: 1943-1993 (San Antonio, TX, 1993), Contemp. Math., Vol. 167, Amer. Math. Soc., Providence, RI, 1994, 240-291.
  15. Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Text Editions, Springer, Berlin, 2004.
  16. Bott R., Seeley R., Some remarks on the paper of Callias: ''Axial anomalies and index theorems on open spaces'', Comm. Math. Phys. 62 (1978), 235-245.
  17. Braverman M., Index theorem for equivariant Dirac operators on noncompact manifolds, $K$-Theory 27 (2002), 61-101.
  18. Braverman M., Maschler G., Equivariant APS index for Dirac operators of non-product type near the boundary, Indiana Univ. Math. J. 68 (2019), 435-501, arXiv:1702.08105.
  19. Bunke U., A $K$-theoretic relative index theorem and Callias-type Dirac operators, Math. Ann. 303 (1995), 241-279.
  20. Callias C., Axial anomalies and index theorems on open spaces, Comm. Math. Phys. 62 (1978), 213-234.
  21. Donnelly H., Eta invariants for $G$-spaces, Indiana Univ. Math. J. 27 (1978), 889-918.
  22. Gilkey P.B., On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math. 102 (1993), 129-183.
  23. Gromov M., Lawson Jr. H.B., Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83-196.
  24. Grubb G., Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems, Comm. Partial Differential Equations 17 (1992), 2031-2077.
  25. Hitchin N., Harmonic spinors, Adv. Math. 14 (1974), 1-55.
  26. Hochs P., Wang B.L., Wang H., An equivariant Atiyah-Patodi-Singer index theorem for proper actions I: The index formula, Int. Math. Res. Not. 2023 (2023), 3138-3193, arXiv:1904.11146.
  27. Hochs P., Wang H., An absolute version of the Gromov-Lawson relative index theorem, arXiv:2110.00376.
  28. Kucerovsky D., A short proof of an index theorem, Proc. Amer. Math. Soc. 129 (2001), 3729-3736.
  29. Levitan B.M., Sargsjan I.S., Sturm-Liouville and Dirac operators, Math. Appl. (Sov. Series), Vol. 59, Kluwer Academic Publishers Group, Dordrecht, 1991.
  30. Lott J., Delocalized $L^2$-invariants, J. Funct. Anal. 169 (1999), 1-31, arXiv:dg-ga/9612003.
  31. Mazzeo R., Phillips R.S., Hodge theory on hyperbolic manifolds, Duke Math. J. 60 (1990), 509-559.
  32. Moroianu S., Fibered cusp versus $d$-index theory, Rend. Semin. Mat. Univ. Padova 117 (2007), 193-203, arXiv:math.DG/0610723.
  33. Moscovici H., $L^{2}$-index of elliptic operators on locally symmetric spaces of finite volume, in Operator Algebras and $K$-theory (San Francisco, Calif., 1981), Contemp. Math., Vol. 10, Amer. Math. Soc., Providence, R.I., 1982, 129-137.
  34. Müller W., Spectral theory for Riemannian manifolds with cusps and a related trace formula, Math. Nachr. 111 (1983), 197-288.
  35. Müller W., Manifolds with cusps of rank one. Spectral theory and $L^2$-index theorem, Lecture Notes in Math., Vol. 1244, Springer, Berlin, 1987.
  36. Nomizu K., Ozeki H., The existence of complete Riemannian metrics, Proc. Amer. Math. Soc. 12 (1961), 889-891.
  37. Reed M., Simon B., Methods of modern mathematical physics. II. Fourier analysis, self-adjointness, Academic Press, New York, 1975.
  38. Salomonsen G., Geometric heat kernel coefficients for APS-type boundary conditions, in Journées ''Équations aux Dérivées Partielles'' (Saint-Jean-de-Monts, 1998), Nantes Université, Nantes, 1998, Exp. No. XI, 19 pages.
  39. Stern M., $L^2$-index theorems on locally symmetric spaces, Invent. Math. 96 (1989), 231-282.
  40. Stern M.A., $L^2$-index theorems on warped products, Ph.D. Thesis, Princeton University, 1984.
  41. Titchmarsh E.C., Eigenfunction expansions associated with second-order differential equations. Part I, 2nd ed., Clarendon Press, Oxford, 1962.
  42. Vaillant B., Index- and spectral theory for manifolds with generalized fibred cusps, Bonner Mathematische Schriften, Vol. 344, Universität Bonn, Bonn, 2001, arXiv:math.DG/0102072.

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