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.
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[BogomolovDeOliveira]
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] 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] 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] 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] 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] 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] 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] 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] 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},
}
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