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The Cauchy-Riemann equations and differential geometry. (English) Zbl 0496.32012

MSC:
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
53C40 Global submanifolds
32-03 History of several complex variables and analytic spaces
01A65 Development of contemporary mathematics
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[1] M. F. Atiyah, Geometry on Yang-Mills fields, Scuola Normale Superiore Pisa, Pisa, 1979. · Zbl 0435.58001
[2] M. S. Baouendi and F. Trèves, A property of the functions and distributions annihilated by a locally integrable system of complex vector fields, Ann. of Math. (2) 113 (1981), no. 2, 387 – 421. · Zbl 0491.35036
[3] Stefan Bergman, The Kernel Function and Conformal Mapping, Mathematical Surveys, No. 5, American Mathematical Society, New York, N. Y., 1950. · Zbl 0040.19001
[4] H. Behnke and P. Thullen, Theorie der Funktionen mehrerer komplexer Veränderlichen, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 51. Zweite, erweiterte Auflage. Herausgegeben von R. Remmert. Unter Mitarbeit von W. Barth, O. Forster, H. Holmann, W. Kaup, H. Kerner, H.-J. Reiffen, G. Scheja und K. Spallek, Springer-Verlag, Berlin-New York, 1970 (German). · Zbl 0204.39502
[5] S. Bochner, Analytic and meromorphic continuation by means of Green’s formula, Ann. of Math. (2) 44 (1943), 652 – 673. · Zbl 0060.24206
[6] L. Boutet de Monvel, Intégration des équations de Cauchy-Riemann induites formelles, Séminaire Goulaouic-Lions-Schwartz 1974 – 1975; Équations aux derivées partielles linéaires et non linéaires, Centre Math., École Polytech., Paris, 1975, pp. Exp. No. 9, 14 (French). · Zbl 0317.58003
[7] Arthur B. Brown, On certain analytic continuations and analytic homeomorphisms, Duke Math. J. 2 (1936), no. 1, 20 – 28. · Zbl 0013.40701
[8] Robert L. Bryant, Holomorphic curves in Lorentzian CR-manifolds, Trans. Amer. Math. Soc. 272 (1982), no. 1, 203 – 221. · Zbl 0517.32008
[9] D. Burns Jr. and S. Shnider, Real hypersurfaces in complex manifolds, Several complex variables (Proc. Sympos. Pure Math., Vol. XXX, Part 2, Williams Coll., Williamstown, Mass., 1975) Amer. Math. Soc., Providence, R.I., 1977, pp. 141 – 168.
[10] D. Burns Jr., S. Shnider, and R. O. Wells Jr., Deformations of strictly pseudoconvex domains, Invent. Math. 46 (1978), no. 3, 237 – 253. · Zbl 0412.32022
[11] C. Carathéodory, Über das Schwarze Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen, Math. Ann. 97 (1926), 76-98. · JFM 52.0345.02
[12] Élie Cartan, Sur la géométrie pseudo-conforme des hypersurfaces de l’espace de deux variables complexes II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (2) 1 (1932), no. 4, 333 – 354 (French). · Zbl 0005.37401
[13] Shiing Shen Chern, On the projective structure of a real hypersurface in \?_{\?+1}, Math. Scand. 36 (1975), 74 – 82. Collection of articles dedicated to Werner Fenchel on his 70th birthday. · Zbl 0305.53019
[14] S. S. Chern and J. K. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219 – 271. · Zbl 0302.32015
[15] Paul R. Dippolito, Universal bundles for deformations of asymmetric structures, Trans. Amer. Math. Soc. 271 (1982), no. 1, 101 – 115. · Zbl 0491.58034
[16] James John Faran, Segre families and real hypersurfaces, Invent. Math. 60 (1980), no. 2, 135 – 172. · Zbl 0464.32011
[17] Charles Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1 – 65. · Zbl 0289.32012
[18] Charles L. Fefferman, Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), no. 2, 395 – 416. , https://doi.org/10.2307/1970945 C. Fefferman, Correction to: ”Monge-Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains” (Ann. of Math. (2) 103 (1976), no. 2, 395 – 416), Ann. of Math. (2) 104 (1976), no. 2, 393 – 394. · Zbl 0332.32018
[19] Charles Fefferman, Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), no. 2, 131 – 262. · Zbl 0444.32013
[20] G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy-Riemann complex, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1972. Annals of Mathematics Studies, No. 75. · Zbl 0247.35093
[21] H. Grauert and R. Remmert, Analytische Stellenalgebren, Springer-Verlag, Berlin-New York, 1971 (German). Unter Mitarbeit von O. Riemenschneider; Die Grundlehren der mathematischen Wissenschaften, Band 176. · Zbl 0231.32001
[22] Robert E. Greene and Steven G. Krantz, Deformation of complex structures, estimates for the \partial equation, and stability of the Bergman kernel, Adv. in Math. 43 (1982), no. 1, 1 – 86. · Zbl 0504.32016
[23] Robert C. Gunning and Hugo Rossi, Analytic functions of several complex variables, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1965. · Zbl 0141.08601
[24] Fritz Hartogs, Zur Theorie der analytischen Funktionen mehrerer unabhängiger Veränderlichen, insbesondere über die Darstellung derselben durch Reihen, welche nach Potenzen einer Veränderlichen fortschreiten, Math. Ann. 62 (1906), no. 1, 1 – 88 (German). · JFM 37.0444.01
[25] Lars Hörmander, Linear partial differential operators, Die Grundlehren der mathematischen Wissenschaften, Bd. 116, Academic Press, Inc., Publishers, New York; Springer-Verlag, Berlin-Göttingen-Heidelberg, 1963. · Zbl 0108.09301
[26] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. · Zbl 0271.32001
[27] David S. Johnson, Biholomorphic equivalence in a class of graph domains, Indiana Univ. Math. J. 29 (1980), no. 3, 341 – 348. · Zbl 0442.32004
[28] A. Krzoska, Über die naturlichen Grenzen der analytischen Funktionen mehrerer Veränderlicher, Dissertation, Greifswald, 1933.28a. C. R. LeBrun, Jr., Spaces of complex geodesics and related structures, Thesis, Oxford University, 1980.
[29] E. E. Levi, Studii sui punti singolari essenziale delle funzioni analitiche di due o più variabili complesse, Annali di Mat. 17 (1909), 61-87. · JFM 41.0487.01
[30] Hans Lewy, On the local character of the solutions of an atypical linear differential equation in three variables and a related theorem for regular functions of two complex variables, Ann. of Math. (2) 64 (1956), 514 – 522. · Zbl 0074.06204
[31] Hans Lewy, On hulls of holomorphy, Comm. Pure Appl. Math. 13 (1960), 587 – 591. · Zbl 0113.06102
[32] Jürgen Moser, Holomorphic equivalence and normal forms of hypersurfaces, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973) Amer. Math. Soc., Providence, R. I., 1975, pp. 109 – 112.
[33] Louis Nirenberg, Lectures on linear partial differential equations, American Mathematical Society, Providence, R.I., 1973. Expository Lectures from the CBMS Regional Conference held at the Texas Technological University, Lubbock, Tex., May 22 – 26, 1972; Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 17. · Zbl 0267.35001
[34] Roger Penrose, Nonlinear gravitons and curved twistor theory, General Relativity and Gravitation 7 (1976), no. 1, 31 – 52. The riddle of gravitation – on the occasion of the 60th birthday of Peter G. Bergmann (Proc. Conf., Syracuse Univ., Syracuse, N. Y., 1975). · Zbl 0354.53025
[35] H. Poincaré, Les functions analytique de deux variables et la représentation conforme, Rend. Circ. Math. Palermo 23 (1907), 185-220 (or Oeuvres. IV, 244-289). · JFM 38.0459.02
[36] W. F. Osgood, Lehrbuch der Funktionentheorie, 2nd. ed., vol. 2, part I, Teubner, Leipzig, 1929. · JFM 55.0171.02
[37] B. Segre, Questioni geometriche legate colla teoria delle funzioni di due variabili complesse, Rend. Sem. Mat. Roma 7 (1931), 59-107. · Zbl 0005.01901
[38] B. Segre, Interno al problema di Poincaré della rappresentazione pseudoconforme, Rend. Acc. Lincei 13 (1931), pp. 676-683. · Zbl 0003.21302
[39] Friedrich Sommer, Komplex-analytische Blätterung reeller Mannigfaltigkeiten im \?\(^{n}\), Math. Ann. 136 (1958), 111 – 133 (German). · Zbl 0092.29902
[40] Noboru Tanaka, On the pseudo-conformal geometry of hypersurfaces of the space of \? complex variables, J. Math. Soc. Japan 14 (1962), 397 – 429. · Zbl 0113.06303
[41] Noboru Tanaka, Graded Lie algebras and geometric structures, Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965) Nippon Hyoronsha, Tokyo, 1966, pp. 147 – 150. · Zbl 0163.43904
[42] Noboru Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131 – 190. · Zbl 0346.32010
[43] M. A. Tresse, Détermination des invariants ponctuels de l’equation différentielle ordinaire du second order y” = w (x, y, y’), S. Hirzel, Leipzig, 1896. · JFM 27.0254.01
[44] S. M. Webster, On the mapping problem for algebraic real hypersurfaces, Invent. Math. 43 (1977), no. 1, 53 – 68. · Zbl 0348.32005
[45] S. M. Webster, On the Moser normal form at a non-umbilic point, Math. Ann. 233 (1978), no. 2, 97 – 102. · Zbl 0358.32013
[46] S. M. Webster, The rigidity of C-R hypersurfaces in a sphere, Indiana Univ. Math. J. 28 (1979), no. 3, 405 – 416. · Zbl 0387.53020
[47] R. O. Wells Jr., Function theory on differentiable submanifolds, Contributions to analysis (a collection of papers dedicated to Lipman Bers), Academic Press, New York, 1974, pp. 407 – 441.
[48] R. O. Wells Jr., Complex manifolds and mathematical physics, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 2, 296 – 336. · Zbl 0444.32014
[49] R. O. Wells Jr., Complex geometry in mathematical physics, Séminaire de Mathématiques Supérieures [Seminar on Higher Mathematics], vol. 78, Presses de l’Université de Montréal, Montreal, Que., 1982. Notes by Robert Pool.
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