Addendum to: Subelliptic $\operatorname{Spin}_{\mathbb{C}}$ Dirac Operators, III

Abstract

We prove the relative index conjecture, which in turn implies that the set of embeddable deformations of a strictly pseudoconvex CR-structure on a compact 3-manifold is closed in the $\mathcal{C}^{\infty}$-topology.

  • [BogomolovDeOliveira] Go to document F. A. Bogomolov and B. de Oliveira, "Stein small deformations of strictly pseudoconvex surfaces," in Birational Algebraic Geometry, Providence, RI: Amer. Math. Soc., 1997, vol. 207, pp. 25-41.
    @incollection {BogomolovDeOliveira, MRKEY = {1462922},
      AUTHOR = {Bogomolov, Fedor A. and de Oliveira, Bruno},
      TITLE = {Stein small deformations of strictly pseudoconvex surfaces},
      BOOKTITLE = {Birational Algebraic Geometry},
      VENUE={{B}altimore, {MD},
      1996},
      SERIES = {Contemp. Math.},
      VOLUME = {207},
      PAGES = {25--41},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {1997},
      MRCLASS = {32G05 (14C25 32F30)},
      MRNUMBER = {1462922},
      MRREVIEWER = {S. Dimiev},
      DOI = {10.1090/conm/207/02717},
      ZBLNUMBER = {0889.32021},
      }
  • [EpsteinRI1] Go to document C. L. Epstein, "A relative index on the space of embeddable CR-structures. I," Ann. of Math., vol. 147, iss. 1, pp. 1-59, 1998.
    @article {EpsteinRI1, MRKEY = {1609455},
      AUTHOR = {Epstein, Charles L.},
      TITLE = {A relative index on the space of embeddable {CR}-structures. {I}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {147},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {1--59},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {32G07 (32F20 32F40 35P15)},
      MRNUMBER = {1609455},
      MRREVIEWER = {John Bland},
      DOI = {10.2307/120982},
      ZBLNUMBER = {0942.32025},
      }
  • [EpsteinRI1half] Go to document C. L. Epstein, "Erratum: A relative index on the space of embeddable CR-structures. I," Ann. of Math., vol. 154, iss. 1, pp. 223-226, 2001.
    @article {EpsteinRI1half, MRKEY = {MR1847595},
      AUTHOR = {Epstein, Charles L.},
      TITLE = {Erratum: {A} relative index on the space of embeddable {CR}-structures. {I}},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {154},
      YEAR = {2001},
      NUMBER = {1},
      PAGES = {223--226},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {32G07 (32V15 32W10 35P15)},
      MRNUMBER = {1847595},
      MRREVIEWER = {John Bland},
      DOI = {10.2307/3062117},
      ZBLNUMBER = {0983.32036},
     }
  • [EpsteinRI2] Go to document C. L. Epstein, "A relative index on the space of embeddable CR-structures. II," Ann. of Math. (2), vol. 147, iss. 1, pp. 61-91, 1998.
    @article {EpsteinRI2, MRKEY = {MR1609451},
      AUTHOR = {Epstein, Charles L.},
      TITLE = {A relative index on the space of embeddable {CR}-structures. {II}},
      JOURNAL = {Ann. of Math. (2)},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {147},
      YEAR = {1998},
      NUMBER = {1},
      PAGES = {61--91},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {32G07 (32F20 32F40 35P15)},
      MRNUMBER = {1609451},
      MRREVIEWER = {John Bland},
      DOI = {10.2307/120983},
      URL = {http://dx.doi.org/10.2307/120983},
      ZBLNUMBER = {0942.32026},
     }
  • [Epstein44] Go to document C. L. Epstein, "Subelliptic ${ Spin}_{\Bbb C}$ Dirac operators. III. The Atiyah-Weinstein conjecture," Ann. of Math., vol. 168, iss. 1, pp. 299-365, 2008.
    @article {Epstein44, MRKEY = {2415404},
      AUTHOR = {Epstein, Charles L.},
      TITLE = {Subelliptic {${\rm Spin}\sb {\Bbb C}$} {D}irac operators. {III}. {T}he {A}tiyah-{W}einstein conjecture},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {168},
      YEAR = {2008},
      NUMBER = {1},
      PAGES = {299--365},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {58J32 (32G07 32V15 32W05 58J20)},
      MRNUMBER = {2415404},
      MRREVIEWER = {Yuri A. Kordyukov},
      DOI = {10.4007/annals.2008.168.299},
      ZBLNUMBER = {1169.32008},
     }
  • [Lempert1] Go to document L. Lempert, "On three-dimensional Cauchy-Riemann manifolds," J. Amer. Math. Soc., vol. 5, iss. 4, pp. 923-969, 1992.
    @article {Lempert1, MRKEY = {1157290},
      AUTHOR = {Lempert, L{á}szl{ó}},
      TITLE = {On three-dimensional {C}auchy-{R}iemann manifolds},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {5},
      YEAR = {1992},
      NUMBER = {4},
      PAGES = {923--969},
      ISSN = {0894-0347},
      MRCLASS = {32F40 (32C16 32G07)},
      MRNUMBER = {1157290},
      MRREVIEWER = {John Bland},
      DOI = {10.2307/2152715},
      ZBLNUMBER = {0781.32014},
      }
  • [Lempert2] Go to document L. Lempert, "Embeddings of three-dimensional Cauchy-Riemann manifolds," Math. Ann., vol. 300, iss. 1, pp. 1-15, 1994.
    @article {Lempert2, MRKEY = {1289827},
      AUTHOR = {Lempert, L{á}szl{ó}},
      TITLE = {Embeddings of three-dimensional {C}auchy-{R}iemann manifolds},
      JOURNAL = {Math. Ann.},
      FJOURNAL = {Mathematische Annalen},
      VOLUME = {300},
      YEAR = {1994},
      NUMBER = {1},
      PAGES = {1--15},
      ISSN = {0025-5831},
      CODEN = {MAANA},
      MRCLASS = {32F40 (32G07)},
      MRNUMBER = {1289827},
      MRREVIEWER = {Roman Dwilewicz},
      DOI = {10.1007/BF01450472},
      ZBLNUMBER = {0817.32009},
     }
  • [Lempert] Go to document L. Lempert, "Algebraic approximations in analytic geometry," Invent. Math., vol. 121, iss. 2, pp. 335-353, 1995.
    @article {Lempert, MRKEY = {1346210},
      AUTHOR = {Lempert, L{á}szl{ó}},
      TITLE = {Algebraic approximations in analytic geometry},
      JOURNAL = {Invent. Math.},
      FJOURNAL = {Inventiones Mathematicae},
      VOLUME = {121},
      YEAR = {1995},
      NUMBER = {2},
      PAGES = {335--353},
      ISSN = {0020-9910},
      CODEN = {INVMBH},
      MRCLASS = {32E05},
      MRNUMBER = {1346210},
      MRREVIEWER = {A. Yu. Rashkovsky},
      DOI = {10.1007/BF01884302},
      ZBLNUMBER = {0837.32008},
      }
  • [stipsicz] Go to document A. I. Stipsicz, "On the geography of Stein fillings of certain 3-manifolds," Michigan Math. J., vol. 51, iss. 2, pp. 327-337, 2003.
    @article {stipsicz, MRKEY = {1992949},
      AUTHOR = {Stipsicz, Andr{á}s I.},
      TITLE = {On the geography of {S}tein fillings of certain 3-manifolds},
      JOURNAL = {Michigan Math. J.},
      FJOURNAL = {The Michigan Mathematical Journal},
      VOLUME = {51},
      YEAR = {2003},
      NUMBER = {2},
      PAGES = {327--337},
      ISSN = {0026-2285},
      MRCLASS = {53D35 (32Q28 57R17 57R57)},
      MRNUMBER = {1992949},
      MRREVIEWER = {John B. Etnyre},
      DOI = {10.1307/mmj/1060013199},
      ZBLNUMBER = {1043.53066},
      }

Authors

Charles L. Epstein

Department of Mathematics
309 South 33rd Street
University of Pennsylvania
Philadelphia, PA 19104-6395