Illinois Journal of Mathematics

Several complex variables and CR geometry

John P. D’Angelo

Full-text: Open access

Abstract

This paper discusses developments in complex analysis and CR geometry in the last forty years related to the Cauchy–Riemann equations, proper holomorphic mappings between balls, and positivity conditions in complex analysis. The paper includes anecdotes about some of the contributors to these developments.

Article information

Source
Illinois J. Math., Volume 56, Number 1 (2012), 7-19.

Dates
First available in Project Euclid: 27 September 2013

Permanent link to this document
https://projecteuclid.org/euclid.ijm/1380287456

Digital Object Identifier
doi:10.1215/ijm/1380287456

Mathematical Reviews number (MathSciNet)
MR3117014

Zentralblatt MATH identifier
1273.32001

Subjects
Primary: 32-02: Research exposition (monographs, survey articles) 32T25: Finite type domains 32T27: Geometric and analytic invariants on weakly pseudoconvex boundaries 32A70: Functional analysis techniques [See mainly 46Exx] 32V99: None of the above, but in this section 32W05: $\overline\partial$ and $\overline\partial$-Neumann operators 32H35: Proper mappings, finiteness theorems 32M25: Complex vector fields 32V15: CR manifolds as boundaries of domains 35N15: $\overline\partial$-Neumann problem and generalizations; formal complexes [See also 32W05, 32W10, 58J10] 12D15: Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) [See also 11Exx] 14P05: Real algebraic sets [See also 12D15, 13J30] 15B57: Hermitian, skew-Hermitian, and related matrices

Citation

D’Angelo, John P. Several complex variables and CR geometry. Illinois J. Math. 56 (2012), no. 1, 7--19. doi:10.1215/ijm/1380287456. https://projecteuclid.org/euclid.ijm/1380287456


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References

  • M. S. Baouendi, P. Ebenfelt and L. Rothschild, Real submanifolds in complex space and their mappings, Princeton Mathematical Series, vol. 47, Princeton University Press, Princeton, NJ, 1999.
  • M. S. Baouendi and X. Huang, Super-rigidity for holomorphic mappings between hyperquadrics with positive signature, J. Differential Geom. 69 (2005), 379–398.
  • M. S. Baouendi, P. Ebenfelt and X. Huang, Holomorphic mappings between hyperquadrics with small signature difference, Amer. J. Math. 133 (2011), 1633–1661.
  • S. Bell, Biholomorphic mappings and the ${\overline \partial}$-problem, Ann. of Math. (2) 114 (1981), 103–113.
  • B. Berndtsson, An introduction to things ${\overline \partial}$, Analysis and algebraic geometry, IAS/Park City Mathematics Series, vol. 17, Amer. Math. Soc., Providence, RI, 2010, pp. 7–76.
  • T. Bloom, D. Catlin, J. D'Angelo and Y.-T. Siu (eds.), Modern methods in complex analysis, Ann. of Math. Stud., Princeton Univ. Press, Princeton, NJ, 1996.
  • H. Boas and E. Straube, Global regularity of the ${\overline \partial}$-Neumann problem: A survey of the $L^2$-Sobolev theory, Several complex variables (M. Schneider and Y. T. Siu, eds.), Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 79–111.
  • H. Boas, S. Fu and E. Straube, The Bergman kernel function: Explicit formulas and zeroes, Proc. Amer. Math. Soc. 127 (1999), 805–811.
  • E. Calabi, Isometric imbedding of complex manifolds, Ann. of Math. (2) 58 (1953), 1–23.
  • D. Catlin, Necessary conditions for subellipticity of the $\overline{ \partial}$-Neumann problem, Ann. of Math. (2) 117 (1983), 147–171.
  • D. Catlin, Boundary invariants of pseudoconvex domains, Ann. of Math. (2) 120 (1984), 529–586.
  • D. Catlin, Subelliptic estimates for the $\overline{\partial}$-Neumann problem on pseudoconvex domains, Ann. of Math. (2) 126 (1987), 131–191.
  • D. Catlin and J. D'Angelo, A stabilization theorem for Hermitian forms and applications to holomorphic mappings, Math. Res. Lett. 3 (1996), 149–166.
  • D. Catlin and J. D'Angelo, Positivity conditions for bihomogeneous polynomials, Math. Res. Lett. 4 (1997), 1–13.
  • D. Catlin and J. D'Angelo, An isometric imbedding theorem for holomorphic bundles, Math. Res. Lett. 6 (1999), 1–18.
  • D. C. Chang, A. Nagel and E. M. Stein, Estimates for the ${\overline \partial}$-Neumann problem in pseudoconvex domains of finite type in $\mathbf{C^2}$, Acta Math. 169 (1992), 153–228.
  • M. Christ and S. Fu, Compactness in the $\overline \partial$-Neumann problem, magnetic Schrödinger operators, and the Aharonov–Bohm effect, Adv. Math. 197 (2005), 1–40.
  • J. P. D'Angelo, Several complex variables and the geometry of real hypersurfaces, CRC Press, Boca Raton, FL, 1992.
  • J. P. D'Angelo, Real hypersurfaces, orders of contact, and applications, Ann. of Math. (2) 115 (1982), 615–637.
  • J. P. D'Angelo, Inequalities from complex analysis, Carus Mathematical Monograph, vol. 28, Mathematics Association of America, Washington, DC, 2002.
  • J. P. D'Angelo, Hermitian analogues of Hilbert's 17-th problem, Adv. Math. 226 (2011), 4607–4637.
  • J. P. D'Angelo, Proper holomorphic mappings, positivity conditions, and isometric imbedding, J. Korean Math. Soc. 40 (2003), 341–371.
  • J. P. D'Angelo, A gentle introduction to points of finite type on real hypersurfaces, Contemp. Math. 332 (2003), 19–36.
  • J. P. D'Angelo, Invariant CR Mappings, Complex analysis: Several complex variables and connections with PDEs and geometry, Trends in Math., Birkhäuser, Basel, 2008, pp. 95–107.
  • J. D'Angelo and J. J. Kohn, Subelliptic estimates and finite type, Several complex variables (M. Schneider and Y. T. Siu, eds.), Math. Sci. Res. Inst. Publ., vol. 37, Cambridge Univ. Press, Cambridge, 1999, pp. 199–232.
  • J. D'Angelo and J. Lebl, Pfister's theorem fails in the Hermitian case, Proc. Amer. Math. Soc. 140 (2012), 1151–1157.
  • J. D'Angelo and D. Lichtblau, Spherical space forms, CR mappings, and proper maps between balls, J. Geom. Anal. 2 (1992), 391–415.
  • J. D'Angelo and M. Putinar, Hermitian complexity of real polynomial ideals, Int. J. Math. 23 (2012), 1250026.
  • J. D'Angelo and J. Tyson, An invitation to Cauchy–Riemann and sub-Riemannian geometry, Notices Amer. Math. Soc. 57 (2010), 208–219.
  • J. D'Angelo and D. Varolin, Positivity conditions for Hermitian symmetric functions, Asian J. Math. 8 (2004), 215–231.
  • J. Faran, Maps from the two-ball to the three-ball, Invent. Math. 68 (1982), 441–475.
  • J. Faran, The linearity of proper holomorphic maps between balls in the low codimension case, J. Differential Geom. 24 (1986), 15–17.
  • C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.
  • G. B. Folland and J. J. Kohn, The Neumann problem for the Cauchy–Riemann complex, Annals of Math. Studies, vol. 75, Princeton University Press, Princeton, 1972.
  • F. Forstneric, Extending proper holomorphic mappings of positive codimension, Invent. Math. 95 (1989), 31–61.
  • S. Fu and E. Straube, Compactness in the ${\overline\partial}$-Neumann problem, Complex Analysis and Geometry (J. D. McNeal, ed.), Ohio State Univ. Math Res. Inst. Publ., vol. 9, de Gruyter, Berlin, 2001, pp. 141–160.
  • D. Grundmeier, Signature pairs for group-invariant Hermitian polynomials, Internat. J. Math. 22 (2011), 311–343.
  • L. Hörmander, An introduction to complex analysis in several variables, Van Nostrand, Princeton, NJ, 1966.
  • X. Huang and S. Ji, On some rigidity problems in Cauchy–Riemann analysis, Proceedings of the international conference on complex geometry and related fields, AMS/IP Stud. Adv. Math., vol. 39, Amer. Math. Soc., Providence, RI, 2007, pp. 89–107.
  • J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds, I, Ann. of Math. (2) 78 (1963), 112–148.
  • J. J. Kohn, Harmonic integrals on strongly pseudo-convex manifolds, II, Ann. of Math. (2) 79 (1964), 450–472.
  • J. J. Kohn, Boundary behavior of $\bar{\partial}$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geom. 6 (1972), 523–542.
  • J. J. Kohn, Subellipticity of the $\bar{\partial}$-Neumann problem on pseudo-convex domains: Sufficient conditions, Acta Math. 142 (1979), 79–122.
  • J. J. Kohn and L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965), 443–492.
  • J. Lebl and D. Lichtblau, Uniqueness of certain polynomials constant on a line, Linear Algebra Appl. 433 (2010), 824–837.
  • E. Løw, Embeddings and proper holomorphic maps of strictly pseudoconvex domains into polydiscs and balls, Math. Z. 190 (1985), 401–410.
  • J. McNeal, Boundary behavior of the Bergman kernel function in $\mathbf{C^2}$, Duke Math. J. 58 (1989), 499–512.
  • J. McNeal, A sufficient condition for compactness of the $\overline\partial$-Neumann operator, J. Funct. Anal. 195 (2002), 190–205.
  • M. Putinar and C. Scheiderer, Sums of Hermitian squares on pseudoconvex boundaries, Math. Res. Lett. 17 (2010), 1047–1053.
  • D. Quillen, On the representation of Hermitian forms as sums of squares, Invent. Math. 5 (1968), 237–242.
  • L. Rothschild and E. Stein, Hypoelliptic differential operators and nilpotent groups, Acta Math. 137 (1976), 247–320.
  • M.C. Shaw, $L^2$ estimates and existence theorems for the Cauchy–Riemann complex, Invent. Math. 82 (1985), 133–150.
  • E. Straube, Lectures on the $L^2$-Sobolev theory of the ${\overline\partial}$-Neumann problem, E.S.I. Lectures in Mathematics and Physics, European Mathematical Society, Zurich, Switzerland, 2011.
  • A. Tumanov, Extending CR-functions on manifolds of finite type to a wedge, Mat. Sb. 136 (1988), 128–139.
  • W. To and S. Yeung, Effective isometric embeddings for certain Hermitian holomorphic line bundles, J. London Math. Soc. (2) 73 (2006), 607–624.
  • D. Varolin, Geometry of Hermitian algebraic functions: Quotients of squared norms, Amer. J. Math. 130 (2008), 291–315.
  • S. M. Webster, On mapping an $n$-ball into an ($n+1$)-ball in complex spaces, Pacific J. Math. 81 (1979), 267–272.