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Cobordism, Relative Indices and Stein Fillings

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Abstract

In this article we build on the framework developed in Ann. Math. 166, 183–214 ([2007]), 166, 723–777 ([2007]), 167, 1–67 ([2008]) to obtain a more complete understanding of the gluing properties for indices of boundary value problems for the Spin ℂ-Dirac operator with sub-elliptic boundary conditions. We extend our analytic results for sub-elliptic boundary value problems for the Spin ℂ-Dirac operator, and gluing results for the indices of these boundary problems to Spin ℂ-manifolds with several pseudoconvex (pseudoconcave) boundary components. These results are applied to study Stein fillability for compact, 3-dimensional, contact manifolds.

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References

  1. Beals, R., Greiner, P.: Calculus on Heisenberg Manifolds. Ann. Math. Stud., vol. 119. Princeton University Press, Princeton (1988)

    MATH  Google Scholar 

  2. Booss-Bavnbek, B., Wojciechowsi, K.P.: Elliptic Boundary Problems for the Dirac Operator. Birkhäuser, Boston (1996)

    Google Scholar 

  3. Boutet de Monvel, L., Guillemin, V.: The Spectral Theory of Toeplitz Operators. Ann. Math. Stud., vol. 99. Princeton University Press, Princeton (1981)

    MATH  Google Scholar 

  4. Dai, X., Zhang, W.: Splitting of the family index. Commun. Math. Phys. 182, 303–318 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  5. Epstein, C.L.: A relative index on the space of embeddable CR-structures I, II. Ann. Math. 147, 1–59, 61–91 (1998)

    Google Scholar 

  6. Epstein, C.L.: Geometric bounds on the relative index. J. Inst. Math. Jussieu 1, 441–465 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  7. Epstein, C.L.: Subelliptic Spinℂ Dirac operators I. Ann. Math. 166, 183–214 (2007)

    MATH  Google Scholar 

  8. Epstein, C.L.: Subelliptic Spinℂ Dirac operators II. Ann. Math. 166, 723–777 (2007)

    MATH  Google Scholar 

  9. Epstein, C.L.: Subelliptic Spinℂ Dirac operators III. Ann. Math. 167, 1–167 (2008)

    Article  Google Scholar 

  10. Epstein, C.L., Henkin, G.M.: Extension of CR-structures for 3-dimensional pseudoconcave manifolds. In: Multidimensional Complex Analysis and Partial Differential Equations, São Carlos (1995), Contemp. Math., vol. 205, pp. 51–67. Am. Math. Soc., Providence (1997)

    Google Scholar 

  11. Epstein, C.L., Henkin, G.M.: Stability of embeddings for pseudoconcave surfaces and their boundaries. Acta Math. 185, 161–237 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  12. Epstein, C.L., Melrose, R.: Contact degree and the index of Fourier integral operators. Math. Res. Lett. 5, 363–381 (1998)

    MathSciNet  MATH  Google Scholar 

  13. Griffiths, P., Harris, J.: Principles of Algebraic Geometry. Wiley-Interscience, New York (1978)

    MATH  Google Scholar 

  14. Gromov, M., Lawson, H.: Positive scalar curvature and the Dirac operator on complete Riemannian manifolds. Publ. Math. IHES 58, 295–408 (1983)

    Google Scholar 

  15. Guillemin, V., Ginzburg, V., Karshon, Y.: Moment Maps, Cobordisms, and Hamiltonian Group Actions. Mathematical Surveys and Monographs, vol. 98. Am. Math. Soc., Providence (2002)

    MATH  Google Scholar 

  16. Hörmander, L.: L 2-estimates and existence theorems for the \(\overline{\partial}\) -operator. Acta Math. 113, 89–152 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kiremidjian, G.: A direct extension method for CR structures. Math. Ann. 242, 1–19 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Lempert, L.: Embeddings of three dimensional Cauchy–Riemann manifolds. Math. Ann. 300, 1–15 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. Stipsicz, A.I.: On the geography of Stein fillings of certain 3-manifolds. Mich. Math. J. 51, 327–337 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  20. Taylor, M.E.: Noncommutative Microlocal Analysis, Part I. Mem. Am. Math. Soc., vol. 313. Am. Math. Soc., Providence (1984)

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Pennsylvania, Philadelphia, USA

    Charles L. Epstein

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  1. Charles L. Epstein
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Correspondence to Charles L. Epstein.

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Dedicated with gratitude and admiration to Gennadi Henkin on the occasion of his 65th birthday.

This material is based upon work supported by the National Science Foundation under Grant No. 0603973, and the Francis J. Carey term chair. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

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Epstein, C.L. Cobordism, Relative Indices and Stein Fillings. J Geom Anal 18, 341–368 (2008). https://doi.org/10.1007/s12220-008-9010-6

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  • DOI: https://doi.org/10.1007/s12220-008-9010-6

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