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The Dirac operator on spaces with conical singularities and positive scalar curvatures


Author: Arthur Weichung Chou
Journal: Trans. Amer. Math. Soc. 289 (1985), 1-40
MSC: Primary 58G10; Secondary 58G05, 58G11, 58G25
DOI: https://doi.org/10.1090/S0002-9947-1985-0779050-8
MathSciNet review: 779050
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Abstract: We study, in the spirit of Jeff Cheeger, the Dirac operator on a space with conical singularities. We obtain a Bochner-type vanishing theorem and prove an index theorem in the singular case. Also, the relationship with manifolds with boundary is explored. In the Appendix two methods of deforming the metric near the boundary are established and applied to obtain several new results on constructing complete metrics with positive scalar curvature.


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  • [1] M. F. Atiyah, R. Bott and V. K. Patodi, On the heat equation and the index theorem, Invent. Math. 19 (1973), 279-330. MR 0650828 (58:31287)
  • [2] M. F. Atiyah, R. Bott and A. A. Shapiro, Clifford modules, Topology 3 (Suppl. 1) (1964), 3-38. MR 0167985 (29:5250)
  • [3] M. F. Atiyah, V. K. Patodi, and I. Singer, Spectral asymmetry and riemannian geometry, I, Math. Proc. Cambridge Philos. Soc. 77 (1975), 43-69. MR 0397797 (53:1655a)
  • [4] M. F. Atiyah, and I. Singer, The index of elliptic operators. III, Ann. of Math. (2) 87 (1968), 546-604. MR 0236952 (38:5245)
  • [5] J. Cheeger, Analytic torsion and the heat equation, Ann. of Math. (2) 109 (1979), 259-322. MR 528965 (80j:58065a)
  • [6] -, On the spectral geometry of spaces with cone-like singularities, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), 2103-2106. (A revised version is in preprint (1980).) MR 530173 (80k:58098)
  • [7] -, On the Hodge theory of riemannian pseudomanifolds, Proc. Sympos. Pure Math., vol. 36, Amer. Math. Soc. Providence, R.I., 1980, pp. 91-145. MR 573430 (83a:58081)
  • [8] -, Spectral geometry of singular riemannian spaces, J. Differential Geom. 18 (1983), 575-657. MR 730920 (85d:58083)
  • [9] J. Cheeger and D. Ebin, Comparison theorems in Riemannian geometry, North-Holland, 1975. MR 0458335 (56:16538)
  • [10] J. Cheeger, M. Goresky and R. MacPherson, $ {L^2}$-cohomology and intersection homology of singular algebraic varieties, Seminar on Differential Geometry, Ann. of Math. Stud., no. 102, Princeton Univ. Press, Princeton, N.J., 1982, pp. 303-340. MR 645745 (84f:58005)
  • [11] J. Cheeger and M. Taylor, On the diffraction of waves by conical singularities. I, Comm. Pure Appl. Math. 35 (1982), 275-331. MR 649347 (84h:35091a)
  • [12] A. Erdelyi (Editor), Higher transcendental functions (Bateman Manuscript Project), Vol. 2, McGraw-Hill, 1953.
  • [13] T. Friedrich, Der erste eigenwert des Dirac-operators einer kompakten riemannschen mannigfaltigkeit nichnegativer skalarkrümmung, Universität zu Berlin (preprint). MR 600828 (82g:58088)
  • [14] K. O. Friedrich, The identity of weak and strong extensions of differential operators, Trans. Amer. Soc. 55 (1944), 132-151. MR 0009701 (5:188b)
  • [15] M. Gaffney, A special Stoke's theorem for riemannian manifolds, Ann. of Math. (2) 60 (1954), 140-145. MR 0062490 (15:986d)
  • [16] M. Gromov and H. B. Lawson, Spin and the scalar curvature in the presence of a fundamental group. I, Ann. of Math (2) 111 (1980), 209-230. MR 569070 (81g:53022)
  • [17] -, The classification of simply connected manifolds of positive scalar curvature, Ann. of Math. (2) 111 (1980), 423-434. MR 577131 (81h:53036)
  • [18] -, Positive scalar curvature and the Dirac operator on complete riemannian manifolds, I.H.E.S. Publ. Math. 58 (1983), 83-196. MR 720933 (85g:58082)
  • [19] E. L. Ince, Ordinary differential equations, Dover, New York, 1956. MR 0010757 (6:65f)
  • [20] P. M. Ingram, Extension aspects and new examples of positively Ricci curved manifolds, Ph.D. Thesis, State Univ. of New York at Stony Brook, Stony Brook, N. Y., 1981.
  • [21] N. Lebedev, Special functions and their applications, Dover, 1972. MR 0350075 (50:2568)
  • [22] H. B. Lawson and M. L. Michelsohn, The geometry of spinors (to appear).
  • [23] A. Lichnerowicz, Spineurs harmoniques, C. R. Acad. Sci. Paris Ser. A-B 257 (1963), 7-9. MR 0156292 (27:6218)
  • [24] J. Milnor, Remarks concerning spin manifolds, Differential and Combinatorial Topology, Princeton Univ. Press, Princeton, N.J., 1964, pp. 55-62. MR 0180978 (31:5208)
  • [25] M. Reed and B. Simon, Functional analysis, methods of modern mathematical physics, Vol. 1, Academic Press, 1972.
  • [26] G. deRham, Variétés différentiables , Hermann, Paris, 1960.
  • [27] I. Sneddon, The use of integral transforms, McGraw-Hill, 1972.
  • [28] I. Stakgold, Boundary value problems of mathematical physics, Vol. 1, Macmillan, 1967. MR 0205776 (34:5602)
  • [29] G. Watson, A treatise on the theory of Bessel functions, Cambridge Univ. Press, 1973. MR 0010746 (6:64a)

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DOI: https://doi.org/10.1090/S0002-9947-1985-0779050-8
Article copyright: © Copyright 1985 American Mathematical Society

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