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

SIGMA 12 (2016), 089, 45 pages      arXiv:1312.4241      https://doi.org/10.3842/SIGMA.2016.089

The Index of Dirac Operators on Incomplete Edge Spaces

Pierre Albin a and Jesse Gell-Redman b
a) University of Illinois, Urbana-Champaign, USA
b) Department of Mathematics, University of Melbourne, Melbourne, Australia

Received November 02, 2015, in final form August 30, 2016; Published online September 08, 2016

Abstract
We derive a formula for the index of a Dirac operator on a compact, even-dimensional incomplete edge space satisfying a ''geometric Witt condition''. We accomplish this by cutting off to a smooth manifold with boundary, applying the Atiyah-Patodi-Singer index theorem, and taking a limit. We deduce corollaries related to the existence of positive scalar curvature metrics on incomplete edge spaces.

Key words: Atiyah-Singer index theorem; Dirac operators; singular spaces; positive scalar curvature.

pdf (724 kb)   tex (69 kb)

References

  1. Abramowitz M., Stegun I.A., Handbook of mathematical functions with formulas, graphs, and mathematical tables, National Bureau of Standards Applied Mathematics Series, Vol. 55, U.S. Government Printing Office, Washington, D.C., 1964.
  2. Albin P., Leichtnam É., Mazzeo R., Piazza P., The signature package on Witt spaces, Ann. Sci. Éc. Norm. Supér. (4) 45 (2012), 241-310, arXiv:1112.0989.
  3. Albin P., Leichtnam É., Mazzeo R., Piazza P., Hodge theory on Cheeger spaces, J. Reine Angew. Math., to appear, arXiv:1307.5473.
  4. Ammann B., Humbert E., Morel B., Mass endomorphism and spinorial Yamabe type problems on conformally flat manifolds, Comm. Anal. Geom. 14 (2006), 163-182, math.DG/0503299.
  5. Ammann B., Lauter R., Nistor V., Pseudodifferential operators on manifolds with a Lie structure at infinity, Ann. of Math. 165 (2007), 717-747, math.AP/0304044.
  6. Amos D.E., Computation of modified Bessel functions and their ratios, Math. Comp. 28 (1974), 239-251.
  7. Atiyah M., Lebrun C., Curvature, cones and characteristic numbers, Math. Proc. Cambridge Philos. Soc. 155 (2013), 13-37, arXiv:1203.6389.
  8. Atiyah M.F., Patodi V.K., Singer I.M., Spectral asymmetry and Riemannian geometry. I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69.
  9. Baricz Á., Bounds for modified Bessel functions of the first and second kinds, Proc. Edinb. Math. Soc. 53 (2010), 575-599.
  10. Baricz Á., Bounds for Turánians of modified Bessel functions, arXiv:1202.4853.
  11. Berline N., Getzler E., Vergne M., Heat kernels and Dirac operators, Grundlehren Text Editions, Springer-Verlag, Berlin, 2004.
  12. Bismut J.M., Cheeger J., $\eta$-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), 33-70.
  13. Bismut J.M., Cheeger J., Families index for manifolds with boundary, superconnections, and cones. I. Families of manifolds with boundary and Dirac operators, J. Funct. Anal. 89 (1990), 313-363.
  14. Bismut J.M., Cheeger J., Families index for manifolds with boundary, superconnections and cones. II. The Chern character, J. Funct. Anal. 90 (1990), 306-354.
  15. Bismut J.M., Cheeger J., Remarks on the index theorem for families of Dirac operators on manifolds with boundary, in Differential Geometry, Pitman Monogr. Surveys Pure Appl. Math., Vol. 52, Longman Sci. Tech., Harlow, 1991, 59-83.
  16. Bismut J.M., Freed D.S., The analysis of elliptic families. II. Dirac operators, eta invariants, and the holonomy theorem, Comm. Math. Phys. 107 (1986), 103-163.
  17. Booß-Bavnbek B., Wojciechowski K.P., Elliptic boundary problems for Dirac operators, Mathematics: Theory \rm & Applications, Birkhäuser Boston, Inc., Boston, MA, 1993.
  18. Bott R., Tu L.W., Differential forms in algebraic topology, Graduate Texts in Mathematics, Vol. 82, Springer-Verlag, New York - Berlin, 1982.
  19. Brüning J., The signature operator on manifolds with a conical singular stratum, Astérisque (2009), 1-44).
  20. Brüning J., Seeley R., An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), 659-714.
  21. Chan S.W., $\mathcal L$-classes on pseudomanifolds with one singular stratum, Proc. Amer. Math. Soc. 125 (1997), 1955-1968.
  22. Cheeger J., On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. USA 76 (1979), 2103-2106.
  23. Cheeger J., On the Hodge theory of Riemannian pseudomanifolds, in Geometry of the Laplace Operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., Vol. 36, Amer. Math. Soc., Providence, R.I., 1980, 91-146.
  24. Cheeger J., Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575-657.
  25. Chou A.W., The Dirac operator on spaces with conical singularities and positive scalar curvatures, Trans. Amer. Math. Soc. 289 (1985), 1-40.
  26. Chou A.W., Criteria for selfadjointness of the Dirac operator on pseudomanifolds, Proc. Amer. Math. Soc. 106 (1989), 1107-1116.
  27. Dai X., Adiabatic limits, nonmultiplicativity of signature, and Leray spectral sequence, J. Amer. Math. Soc. 4 (1991), 265-321.
  28. Dai X., Wei G., Hitchin-Thorpe inequality for noncompact Einstein 4-manifolds, Adv. Math. 214 (2007), 551-570, math.DG/0612105.
  29. Dai X., Zhang W.P., Circle bundles and the Kreck-Stolz invariant, Trans. Amer. Math. Soc. 347 (1995), 3587-3593.
  30. Debord C., Lescure J.M., Nistor V., Groupoids and an index theorem for conical pseudo-manifolds, J. Reine Angew. Math. 628 (2009), 1-35, math.OA/0609438.
  31. Fedosov B., Schulze B.W., Tarkhanov N., The index of elliptic operators on manifolds with conical points, Selecta Math. (N.S.) 5 (1999), 467-506.
  32. Gil J.B., Loya P.A., Mendoza G.A., A note on the index of cone differential operators, math.AP/0110172.
  33. Gil J.B., Mendoza G.A., Adjoints of elliptic cone operators, Amer. J. Math. 125 (2003), 357-408, math.AP/0108095.
  34. Gilkey P.B., On the index of geometrical operators for Riemannian manifolds with boundary, Adv. Math. 102 (1993), 129-183.
  35. Grubb G., Heat operator trace expansions and index for general Atiyah-Patodi-Singer boundary problems, Comm. Partial Differential Equations 17 (1992), 2031-2077.
  36. Hausel T., Hunsicker E., Mazzeo R., Hodge cohomology of gravitational instantons, Duke Math. J. 122 (2004), 485-548, math.DG/0207169.
  37. Hörmander L., The analysis of linear partial differential operators. III. Pseudo-differential operators, Classics in Mathematics, Springer, Berlin, 2007.
  38. Krainer T., Mendoza G.A., Boundary value problems for elliptic wedge operators: the first-order case, in Elliptic and Parabolic Equations, Springer Proc. Math. Stat., Vol. 119, Springer, Cham, 2015, 209-232, arXiv:1403.6894.
  39. Kronheimer P.B., Mrowka T.S., Gauge theory for embedded surfaces. I, Topology 32 (1993), 773-826.
  40. Kronheimer P.B., Mrowka T.S., Gauge theory for embedded surfaces. II, Topology 34 (1995), 37-97.
  41. Lawson Jr. H.B., Michelsohn M.L., Spin geometry, Princeton Mathematical Series, Vol. 38, Princeton University Press, Princeton, NJ, 1989.
  42. Leichtnam E., Mazzeo R., Piazza P., The index of Dirac operators on manifolds with fibered boundaries, Bull. Belg. Math. Soc. Simon Stevin 13 (2006), 845-855, math.DG/0609614.
  43. Lesch M., Operators of Fuchs type, conical singularities, and asymptotic methods, Teubner-Texte zur Mathematik, Vol. 136, B.G. Teubner Verlagsgesellschaft mbH, Stuttgart, 1997, dg-ga/9607005.
  44. Lock M.T., Viaclovsky J.A., An index theorem for anti-self-dual orbifold-cone metrics, Adv. Math. 248 (2013), 698-716, arXiv:1209.3243.
  45. Mazzeo R., Elliptic theory of differential edge operators. I, Comm. Partial Differential Equations 16 (1991), 1615-1664.
  46. Mazzeo R., Melrose R.B., The adiabatic limit, Hodge cohomology and Leray's spectral sequence for a fibration, J. Differential Geom. 31 (1990), 185-213.
  47. Mazzeo R., Vertman B., Analytic torsion on manifolds with edges, Adv. Math. 231 (2012), 1000-1040, arXiv:1103.0448.
  48. Mazzeo R., Vertman B., Elliptic theory of differential edge operators, II: Boundary value problems, Indiana Univ. Math. J. 63 (2014), 1911-1955, arXiv:1307.2266.
  49. Melrose R.B., Pseudodifferential operators, corners and singular limits, in Proceedings of the International Congress of Mathematicians, Vols. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 217-234.
  50. Melrose R.B., Calculus of conormal distributions on manifolds with corners, Int. Math. Res. Not. 1992 (1992), 51-61.
  51. Melrose R.B., The Atiyah-Patodi-Singer index theorem, Research Notes in Mathematics, Vol. 4, A K Peters, Ltd., Wellesley, MA, 1993.
  52. Melrose R.B., Fibrations, compactifications and algebras of pseudodifferential operators, in Partial Differential Equations and Mathematical Physics (Copenhagen, 1995; Lund, 1995), Progr. Nonlinear Differential Equations Appl., Vol. 21, Birkhäuser Boston, Boston, MA, 1996, 246-261.
  53. Melrose R.B., Differential analysis on manifolds with corners, available at http://www-math.mit.edu/~rbm/book.html.
  54. Melrose R.B., Introduction to microlocal analysis, available at http://www-math.mit.edu/~rbm/Lecture_notes.html.
  55. Melrose R.B., Nistor V., Homology of pseudodifferential operators. I. Manifolds with boundary, funct-an/9606005.
  56. Nistor V., Analysis on singular spaces: Lie manifolds and operator algebras, J. Geom. Phys. 105 (2016), 75-101, arXiv:1512.06575.
  57. Penfold R., Vanden-Broeck J.M., Grandison S., Monotonicity of some modified Bessel function products, Integral Transforms Spec. Funct. 18 (2007), 139-144.
  58. Roe J., Elliptic operators, topology and asymptotic methods, Pitman Research Notes in Mathematics Series, Vol. 395, 2nd ed., Longman, Harlow, 1998.
  59. Seeley R.T., Complex powers of an elliptic operator, in Singular Integrals (Proc. Sympos. Pure Math., Chicago, Ill., 1966), Amer. Math. Soc., Providence, R.I., 1967, 288-307.
  60. Taylor M.E., Partial differential equations. II. Basic theory, Applied Mathematical Sciences, Vol. 115, 2nd ed., Springer, New York, 2011,.
  61. Taylor M.E., Partial differential equations. II. Qualitative studies of linear equations, Applied Mathematical Sciences, Vol. 116, 2nd ed., Springer, New York, 2011.
  62. Witten E., Global gravitational anomalies, Comm. Math. Phys. 100 (1985), 197-229.

Previous article  Next article   Contents of Volume 12 (2016)