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Index theory for perturbed Dirac operators on manifolds with conical singularities


Authors: Jeffrey Fox and Peter Haskell
Journal: Proc. Amer. Math. Soc. 123 (1995), 2265-2273
MSC: Primary 58G12; Secondary 47A53, 47F05, 57R15, 57S25
DOI: https://doi.org/10.1090/S0002-9939-1995-1243166-4
MathSciNet review: 1243166
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Abstract | References | Similar Articles | Additional Information

Abstract: On an odd-dimensional manifold with isolated conical singularities, we perturb a Dirac operator by a vector bundle endomorphism whose pointwise norm grows in inverse proportion to the distance from the singular set. We give two proofs of an index formula for the resulting Fredholm operator. We mention an application to the index theory of transversally elliptic operators.


References [Enhancements On Off] (What's this?)

  • [A] N. Anghel, $ {L^2}$-index formulae for perturbed Dirac operators, Comm. Math. Phys. 128 (1990), 77-97. MR 1042444 (91b:58243)
  • [At] M. F. Atiyah, Elliptic operators and compact groups, Lecture Notes in Math., vol. 401, Springer-Verlag, Berlin, 1974. MR 0482866 (58:2910)
  • [BiC] J.-M. Bismut and J. Cheeger, $ \eta $-invariants and their adiabatic limits, J. Amer. Math. Soc. 2 (1989), 33-70. MR 966608 (89k:58269)
  • [BoS] R. Bott and R. Seeley, Some remarks on the paper of Callias, Comm. Math. Phys. 62 (1978), 235-245. MR 507781 (80h:58045b)
  • [B1] J. Brüning, $ {L^2}$-index theorems on certain complete manifolds, J. Differential Geom. 32 (1990), 491-532. MR 1072916 (91h:58103)
  • [B2] -, On $ {L^2}$-index theorems for complete manifolds of rank-one type, Duke Math. J. 66 (1992), 257-309. MR 1162191 (93i:58145)
  • [BHe1] J. Brüning and E. Heintze, The asymptotic expansion of Minakshisundaram-Pleijel in the equivariant case, Duke Math. J. 51 (1984), 959-980. MR 771390 (86b:58124)
  • [BHe2] -, Representations of compact Lie groups and elliptic operators, Invent. Math. 50 (1979), 169-203. MR 517776 (81b:58039)
  • [BM] J. Brüning and H. Moscovici, $ {L^2}$-index for certain Dirac-Schrödinger operators, Duke Math. J. 66 (1992), 311-336. MR 1162192 (93g:58142)
  • [BS1] J. Brüning and R. Seeley, An index theorem for first order regular singular operators, Amer. J. Math. 110 (1988), 659-714. MR 955293 (89k:58271)
  • [BS2] -, The resolvent expansion for second order regular singular operators, J. Funct. Anal. 73 (1987), 369-429. MR 899656 (88g:35151)
  • [Ca] C. Callias, Axial anomalies and index theorems on open spaces, Comm. Math. Phys. 62 (1978), 213-234. MR 507780 (80h:58045a)
  • [C1] J. Cheeger, $ \eta $-invariants, the adiabatic approximation and conical singularities, J. Differential Geom. 26 (1987), 175-221. MR 892036 (89c:58123)
  • [C2] -, On the Hodge theory of Riemannian pseudomanifolds, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc., Providence, RI, 1980, pp. 91-146. MR 573430 (83a:58081)
  • [C3] -, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575-657. MR 730920 (85d:58083)
  • [C4] -, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103-2106. MR 530173 (80k:58098)
  • [Ch1] A. Chou, Criteria for self-adjointness of the Dirac operator on pseudomanifolds, Proc. Amer. Math. Soc. 106 (1989), 1107-1116. MR 975634 (90k:58229)
  • [Ch2] -, The Dirac operator on spaces with conical singularities and positive scalar curvatures, Trans. Amer. Math. Soc. 289 (1985), 1-40. MR 779050 (86i:58124)
  • [Ga] C. Gajdzinski, $ {L^2}$-index for perturbed Dirac operator on odd dimensional open complete manifold, preprint, 1993.
  • [GLa] M. Gromov and H. B. Lawson, Positive scalar curvature and the Dirac operator on complete Riemannian manifolds, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 83-196. MR 720933 (85g:58082)
  • [LaMi] H. B. Lawson and M.-L. Michelsohn, Spin geometry, Princeton Univ. Press, Princeton, NJ, 1989. MR 1031992 (91g:53001)
  • [L] M. Lesch, Deficiency indices for symmetric Dirac operators on manifolds with conic singularities, Topology 32 (1993), 611-623. MR 1231967 (94e:58133)
  • [ReSi] M. Reed and B. Simon, Fourier analysis, self-adjointness, methods of modern mathematical physics, Vol. II, Academic Press, Orlando, FL, 1975.
  • [Sin] I. M. Singer, Recent applications of index theory for elliptic operators, Proc. Sympos. Pure Math., vol. 23, Amer. Math. Soc., Providence, RI, 1973, pp. 11-31. MR 0341538 (49:6286)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1995-1243166-4
Keywords: Perturbed Dirac operator, regular singular operator, relative index theory, adiabatic limit of reduced eta invariants, transversally elliptic operator
Article copyright: © Copyright 1995 American Mathematical Society

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