Abstract
We begin with a little introductory material on the scaling method. Then we use these ideas to discuss Klembeck’s theorem about the boundary asymptotics of the curvature of the Bergman metric on a strictly pseudoconvex domain.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In some geometric contexts this technique is also known as the “method of flattening.” We thank M. Gromov for this comment.
Bibliography
R. Adams, Sobolev Spaces, Academic Press, 1975.
P. Ahern, M. Flores, and W. Rudin, An invariant volume-mean-value property, Jour. Functional Analysis 11(1993), 380–397.
L. Ahlfors, Complex Analysis, 3rd ed., McGraw-Hill, New York, 1979.
N. Aronszajn, Theory of reproducing kernels, Trans. Am. Math. Soc. 68(1950), 337–404.
T. N. Bailey, M. G. Eastwood, and C. R. Graham, Invariant theory for conformal and CR geometry, Annals of Math. 139(1994), 491–552.
D. Barrett, Irregularity of the Bergman projection on a smooth bounded domain in \({\mathbb{C}}^{2},\) Annals of Math. 119(1984), 431–436.
D. Barrett, The behavior of the Bergman projection on the Diederich–Fornaess worm, Acta Math., 168(1992), 1–10.
D. Barrett, Regularity of the Bergman projection and local geometry of domains, Duke Math. Jour. 53(1986), 333–343.
D. Barrett, Behavior of the Bergman projection on the Diederich–Fornæss worm, Acta Math. 168(1992), 1–10.
E. Bedford and P. Federbush, Pluriharmonic boundary values, Tohoku Math. Jour. 26(1974), 505–511.
E. Bedford and J. E. Fornæss, A construction of peak functions on weakly pseudoconvex domains, Ann. Math. 107(1978), 555–568.
E. Bedford and J. E. Fornæss, Counterexamples to regularity for the complex Monge–Ampère equation, Invent. Math. 50 (1978/79), 129–134.
S. Bell, Biholomorphic mappings and the \(\overline{\partial }\) problem, Ann. Math., 114(1981), 103–113.
S. Bell, Local boundary behavior of proper holomorphic mappings, Proc. Sympos. Pure Math, vol. 41, American Math. Soc., Providence R.I., 1984, 1–7.
S. Bell, Differentiability of the Bergman kernel and pseudo-local estimates, Math. Z. 192(1986), 467–472.
S. Bell and H. Boas, Regularity of the Bergman projection in weakly pseudoconvex domains, Math. Annalen 257(1981), 23–30.
S. Bell and D. Catlin, Proper holomorphic mappings extend smoothly to the boundary, Bull. Amer. Math. Soc. (N.S.) 7(1982), 269–272.
S. Bell and S. G. Krantz, Smoothness to the boundary of conformal maps, Rocky Mt. Jour. Math. 17(1987), 23–40.
S. Bell and E. Ligocka, A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math. 57(1980), 283–289.
F. A. Berezin, Quantization in complex symmetric spaces, Math. USSR Izvestia 9(1975), 341–379.
S. Bergman, Über die Entwicklung der harmonischen Funktionen der Ebene und des Raumes nach Orthogonal funktionen, Math. Annalen 86(1922), 238–271.
S. Bergman, The Kernel Function and Conformal Mapping, Am. Math. Soc., Providence, RI, 1970.
S. Bergman and M. Schiffer, Kernel Functions and Elliptic Differential Equations in Mathematical Physics, Academic Press, New York, 1953.
B. Berndtsson, P. Charpentier, A Sobolev mapping property of the Bergman kernel, Math. Z. 235 (2000), 1–10.
L. Bers, Introduction to Several Complex Variables, New York Univ. Press, New York, 1964.
B. E. Blank and S. G. Krantz, Calculus, Key Press, Emeryville, CA, 2006.
Z. Blocki and P. Pflug, Hyperconvexity and Bergman completeness, Nagoya Math. J. 151(1998), 221–225.
T. Bloom and I. Graham, A geometric characterization of points of type m on real submanifolds of \({\mathbb{C}}^{n},\) J. Diff. Geom. 12(1977), 171–182.
H. Boas, Counterexample to the Lu Qi-Keng conjecture, Proc. Am. Math. Soc. 97(1986), 374–375.
H. Boas, The Lu Qi-Keng conjecture fails generically, Proc. Amer. Math. Soc. 124(1996), 2021–2027.
H. Boas and E. Straube, Sobolev estimates for the \(\overline{\partial }\)-Neumann operator on domains in \({\mathbb{C}}^{n}\) admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1991), 81–88.
H. Boas and E. Straube, Equivalence of regularity for the Bergman projection and the \(\overline{\partial }\)-Neumann operator, Manuscripta Math. 67(1990), 25–33.
H. Boas, S. G. Krantz, and M. M. Peloso, unpublished.
S. Bochner, Orthogonal systems of analytic functions, Math. Z. 14(1922), 180–207.
L. Boutet de Monvel, Le noyau de Bergman en dimension 2, Séminaire sur les Équations aux Dérivées Partielles 1987–1988, Exp. no. XXII, École Polytechnique Palaiseau, 1988, p. 13.
L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et Szegő, Soc. Mat. de France Asterisque 34–35(1976), 123–164.
H. J. Bremermann, Holomorphic continuation of the kernel function and the Bergman metric in several complex variables, Lectures on Functions of a Complex Variable, Michigan, 1955, 349–383.
L. Bungart, Holomorphic functions with values in locally convex spaces and applications to integral formulas, Trans. Am. Math. Soc. 111(1964), 317–344.
D. Burns, S. Shnider, R. O. Wells, On deformations of strictly pseudoconvex domains, Invent. Math. 46(1978), 237–253.
L. Caffarelli, J. J. Kohn, L. Nirenberg, and J. Spruck, The Dirichlet problem for nonlinear second-order elliptic equations. II. Complex Monge–Ampère, and uniformly elliptic, equations, Comm. Pure Appl. Math. 38(1985), 209–252.
L. Carleson, Selected Problems on Exceptional Sets, Van Nostrand, Princeton, NJ, 1967.
G. Carrier, M. Crook, and C. Pearson, Functions of a Complex Variable, McGraw-Hill, New York, 1966.
D. Catlin, Necessary conditions for subellipticity of the \(\overline{\partial }-\)Neumann problem, Ann. Math. 117(1983), 147–172.
D. Catlin, Subelliptic estimates for the \(\overline{\partial }\)Neumann problem, Ann. Math. 126(1987), 131–192.
D. Catlin, Boundary behavior of holomorphic functions on pseudoconvex domains, J. Differential Geom. 15(1980), 605–625.
D.-C. Chang, A. Nagel, and E. M. Stein, Estimates for the \(\overline{\partial }\)-Neumann problem in pseudoconvex domains of finite type in \({\mathbb{C}}^{2}\), Acta Math. 169(1992), 153–228.
S.-C. Chen, A counterexample to the differentiability of the Bergman kernel function, Proc. AMS 124(1996), 1807–1810.
B.-Y. Chen and S. Fu, Comparison of the Bergman and Szegö kernels, Advances in Math. 228(2011), 2366–2384.
S.-C. Chen and M. C. Shaw, Partial Differential Equations in Several Complex Variables, AMS/IP Studies in Advanced Mathematics, 19. American Mathematical Society, Providence, RI; International Press, Boston, MA, 2001.
S.-Y. Cheng, Open problems, Conference on Nonlinear Problems in Geometry Held in Katata, September, 1979, Tohoku University, Dept. of Mathematics, Sendai, 1979, p. 2.
S. S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133(1974), 219–271.
M. Christ, Global C ∞ irregularity of the \(\overline{\partial }\)?–Neumann problem for worm domains, J. Amer. Math. Soc. 9(1996), 1171–1185.
M. Christ, Remarks on global irregularity in the \(\overline{\partial }\)-Neumann problem, Several complex variables (Berkeley, CA, 1995–1996), 161–198, Math. Sci. Res. Inst. Publ. 37, Cambridge Univ. Press, Cambridge, 1999.
E. Coddington and N. Levinson, Theory of Ordinary Differential Equations, McGraw-Hill, New York, 1955.
R. R. Coifman and G. Weiss, Analyse Harmonique Non-Commutative sur Certains Espaces Homogenes, Springer Lecture Notes vol. 242, Springer Verlag, Berlin, 1971.
R. Courant and D. Hilbert, Methods of Mathematical Physics, 2nd ed., Interscience, New York, 1966.
J. P. D’Angelo, Real hypersurfaces, orders of contact, and applications, Annals of Math. 115(1982), 615–637.
J. P. D’Angelo, Intersection theory and the \(\overline{\partial }\)- Neumann problem, Proc. Symp. Pure Math. 41(1984), 51–58.
J. P. D’Angelo, Finite type conditions for real hypersurfaces in \({\mathbb{C}}^{n},\) in Complex Analysis Seminar, Springer Lecture Notes vol. 1268, Springer Verlag, 1987, 83–102.
J. P. D’Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, FL, 1993.
K. Diederich, Das Randverhalten der Bergmanschen Kernfunktion und Metrik in streng pseudo-konvexen Gebieten, Math. Ann. 187(1970), 9–36.
K. Diederich, Über die 1. and 2. Ableitungen der Bergmanschen Kernfunktion und ihr Randverhalten, Math. Ann. 203(1973), 129–170.
K. Diederich and J. E. Fornæss, Pseudoconvex domains: An example with nontrivial Nebenhülle, Math. Ann. 225(1977), 275–292.
K. Diederich and J. E. Fornæss, Pseudoconvex domains with real-analytic boundary, Annals of Math. 107(1978), 371–384.
K. Diederich and J. E. Fornæss, Pseudoconvex domains: Bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129–141.
K. Diederich and J. E. Fornæss, Smooth extendability of proper holomorphic mappings, Bull. Amer. Math. Soc. (N.S.) 7(1982), 264–268.
Ebin, D. G., On the space of Riemannian metrics, Bull. Amer. Math. Soc. 74(1968), 1001–1003.
Ebin, D. G., The manifold of Riemannian metrics, 1970 Global Analysis (Proc. Sympos. Pure Math., Vol. XV, Berkeley, Calif., 1968), pp. 11–40, Amer. Math. Soc., Providence, R.I.
M. Engliš, Functions invariant under the Berezin transform, J. Funct. Anal. 121(1994), 233–254.
M. Engliš, Asymptotics of the Berezin transform and quantization on planar domains, Duke Math. J. 79(1995), 57–76.
B. Epstein, Orthogonal Families of Functions, Macmillan, New York, 1965.
A. Erdelyi, et al, Higher Transcendental Functions, McGraw-Hill, New York, 1953.
H. Federer, Geometric Measure Theory, Springer-Verlag, New York, 1969.
C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26(1974), 1–65.
C. Fefferman, Parabolic invariant theory in complex analysis, Adv. Math. 31(1979), 131–262.
G. B. Folland and J. J. Kohn, The Neumann Problem for the Cauchy-Riemann Complex, Princeton University Press, Princeton, NJ, 1972.
G. B. Folland, Spherical harmonic expansion of the Poisson–Szegő kernel for the ball, Proc. Am. Math. Soc. 47(1975), 401–408.
J. E. Fornæss and J. McNeal, A construction of peak functions on some finite type domains. Amer. J. Math. 116(1994), no. 3, 737–755.
F. Forstneric, An elementary proof of Fefferman’s theorem, Expositiones Math., 10(1992), 136–149.
B. Fridman, A universal exhausting domain, Proc. Am. Math. Soc. 98(1986), 267–270.
S. Fu and B. Wong, On strictly pseudoconvex domains with Kähler–Einstein Bergman metrics, Math. Res. Letters 4(1997), 697–703.
T. Gamelin, Uniform Algebras, Prentice-Hall, Englewood Cliffs, NJ, 1969.
T. Gamelin and N. Sibony, Subharmonicity for uniform algebras. J. Funct. Anal. 35 (1980), 64–108.
J. B. Garnett, Analytic Capacity and Measure, Lecture Notes in Math. 297, Springer, New York, 1972.
P. R. Garabedian, A Green’s function in the theory of functions of several complex variables, Ann. of Math. 55(1952). 19–33.
J. Garnett, Bounded Analytic Functions, Academic Press, New York, 1981.
A. Gleason, The abstract theorem of Cauchy-Weil, Pac. J. Math. 12(1962), 511–525.
Goluzin, Geometric Theory of Functions of a Complex Variable, American Mathematical Society, Providence, 1969.
C. R. Graham, The Dirichlet problem for the Bergman Laplacian I, Comm. Partial Diff. Eqs. 8(1983), 433–476.
C. R. Graham, The Dirichlet problem for the Bergman Laplacian II, Comm. Partial Diff. Eqs. 8(1983), 563–641.
C. R. Graham, Scalar boundary invariants and the Bergman kernel, Complex analysis, II (College Park, Md., 1985–86), 108–135, Lecture Notes in Math. 1276, Springer, Berlin, 1987.
C. R. Graham and J. M. Lee, Smooth solutions of degenerate Laplacians on strictly pseudoconvex domains, Duke Jour. Math. 57(1988), 697–720.
I. Graham, Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in \({\mathbb{C}}^{n}\) with smooth boundary, Trans. Am. Math. Soc. 207(1975), 219–240.
H. Grauert and I. Lieb, Das Ramirezsche Integral und die Gleichung \(\overline{\partial }u =\alpha\) im Bereich der beschränkten Formen, Rice University Studies 56(1970), 29–50.
R. E. Greene, K.-T. Kim, and S. G. Krantz, The Geometry of Complex Domains, Birkhäuser Publishing, Boston, MA, 2011.
R. E. Greene and S. G. Krantz, Stability properties of the Bergman kernel and curvature properties of bounded domains, Recent Progress in Several Complex Variables, Princeton University Press, Princeton, 1982.
R. E. Greene and S. G. Krantz, Deformation of complex structures, estimates for the \(\overline{\partial }\) equation, and stability of the Bergman kernel, Adv. Math. 43(1982), 1–86.
R. E. Greene and S. G. Krantz, The automorphism groups of strongly pseudoconvex domains, Math. Annalen 261(1982), 425–446.
R. E. Greene and S. G. Krantz, The stability of the Bergman kernel and the geometry of the Bergman metric, Bull. Am. Math. Soc. 4(1981), 111–115.
R. E. Greene and S. G. Krantz, Stability of the Carathéodory and Kobayashi metrics and applications to biholomorphic mappings, Proc. Symp. in Pure Math., Vol. 41 (1984), 77–93.
R. E. Greene and S. G. Krantz, Normal families and the semicontinuity of isometry and automorphism groups, Math. Zeitschrift 190(1985), 455–467.
R. E. Greene and S. G. Krantz, Characterizations of certain weakly pseudo-convex domains with non-compact automorphism groups, in Complex Analysis Seminar, Springer Lecture Notes 1268(1987), 121–157.
R. E. Greene and S. G. Krantz, Characterization of complex manifolds by the isotropy subgroups of their automorphism groups, Indiana Univ. Math. J. 34(1985), 865–879.
R. E. Greene and S. G. Krantz, Biholomorphic self-maps of domains, Complex Analysis II (C. Berenstein, ed.), Springer Lecture Notes, vol. 1276, 1987, 136–207.
R. E. Greene and S. G. Krantz, Techniques for Studying the automorphism Groups of Weakly Pseudoconvex Domains, Several Complex Variables (Stockholm, 1987/1988), 389–410, Math. Notes, 38, Princeton Univ. Press, Princeton, NJ, 1993.
R. E. Greene and S. G. Krantz, Invariants of Bergman geometry and results concerning the automorphism groups of domains in \({\mathbb{C}}^{n},\) Proceedings of the 1989 Conference in Cetraro (D. Struppa, ed.), to appear.
R. E. Greene and S. G. Krantz, Function Theory of One Complex Variable, 3rd ed., American Mathematical Society, Providence, RI, 2006.
R. Harvey and J. Polking, Fundamental solutions in complex analysis. I. The Cauchy-Riemann operator, Duke Math. J. 46(1979), 253–300.
R. Harvey and J. Polking, Fundamental solutions in complex analysis. II. The induced Cauchy-Riemann operator, Duke Math. J. 46(1979), 301–340.
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962.
G. M. Henkin, Integral representations of functions holomorphic in strictly pseudoconvex domains and some applications, Mat. Sb. 78(120)(1969), 611–632.
E. Hille, Analytic Function Theory, 2nd ed., Ginn and Co., Boston, 1973.
K. Hirachi, The second variation of the Bergman kernel of ellipsoids, Osaka J. Math. 30(1993), 457–473.
K. Hirachi, Scalar pseudo-Hermitian invariants and the Szegő kernel on three-dimensional CR manifolds, Complex Geometry (Osaka, 1990), Lecture Notes Pure Appl. Math., v. 143, Marcel Dekker, New York, 1993, 67–76.
K. Hirachi, Construction of boundary invariants and the logarithmic singularity in the Bergman kernel, Annals of Math. 151(2000), 151–190.
M. Hirsch, Differential Topology, Springer-Verlag, New York, 1976.
L. Hörmander, L 2 estimates and existence theorems for the \(\overline{\partial }\) operator, Acta Math. 113(1965), 89–152.
L. Hörmander, Linear Partial Differential Operators, Springer-Verlag, New York, 1963.
L. Hörmander, Pseudo-differential operators and non-elliptic boundary problems, Ann. Math. 83(1966), 129–209.
L. Hörmander, “Fourier integral operators,” The Analysis of Linear Partial Differential Operators IV, Reprint of the 1994 ed., Springer, Berlin, Heidelberg, New York, 2009.
L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence, 1963.
A. Isaev and S. G. Krantz, Domains with non-compact automorphism group: A Survey, Advances in Math. 146 (1999), 1–38.
S. Jakobsson, Weighted Bergman kernels and biharmonic Green functions, Ph.D. thesis, Lunds Universitet, 2000, 134 pages.
Y. Katznelson, Introduction to Harmonic Analysis, John Wiley and Sons, New York, 1968.
O. Kellogg, Foundations of Potential Theory, Dover, New York, 1953.
N. Kerzman, Hölder and L p estimates for solutions of \(\overline{\partial }u = f\) on strongly pseudoconvex domains, Comm. Pure Appl. Math. 24(1971), 301–380.
N. Kerzman, The Bergman kernel function. Differentiability at the boundary, Math. Ann. 195(1972), 149–158.
N. Kerzman, A Monge–Ampre equation in complex analysis. Several Complex Variables (Proc. Sympos. Pure Math., Vol. XXX, Part 1, Williams Coll., Williamstown, Mass., 1975), pp. 161–167. Amer. Math. Soc., Providence, R.I., 1977.
Y. W. Kim, Semicontinuity of compact group actions on com- pact dierentiable manifolds, Arch. Math. 49(1987), 450–455.
C. Kiselman, A study of the Bergman projection in certain Hartogs domains, Proc. Symposia Pure Math., vol. 52 (E. Bedford, J. D’Angelo, R. Greene, and S. Krantz eds.), American Mathematical Society, Providence, 1991.
P. Klembeck, Kähler metrics of negative curvature, the Bergman metric near the boundary and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ. Math. J. 27(1978), 275–282.
S. Kobayashi, Geometry of bounded domains, Trans. AMS 92(1959), 267–290.
S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Dekker, New York, 1970.
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vols. I and II, Interscience, New York, 1963, 1969.
J. J. Kohn, Quantitative estimates for global regularity, Analysis and geometry in several complex variables (Katata, 1997), 97–128, Trends Math., Birkhäuser Boston, Boston, MA, 1999.
J. J. Kohn, Boundary behavior of \(\overline{\partial }\) on weakly pseudoconvex manifolds of dimension two, J. Diff. Geom. 6(1972), 523–542.
A. Koranyi, Harmonic functions on Hermitian hyperbolic space, Trans. A. M. S. 135(1969), 507–516.
A. Koranyi, Boundary behavior of Poisson integrals on symmetric spaces, Trans. A.M.S. 140(1969), 393–409.
S. G. Krantz, Function Theory of Several Complex Variables, 2nd ed., American Mathematical Society, Providence, RI, 2001.
S. G. Krantz, On a construction of L. Hua for positive reproducing kernels, Michigan Journal of Mathematics 59(2010), 211–230.
S. G. Krantz, Boundary decomposition of the Bergman kernel, Rocky Mountain Journal of Math., to appear.
S. G. Krantz, Partial Differential Equations and Complex Analysis, CRC Press, Boca Raton, FL, 1992.
S. G. Krantz, Cornerstones of Geometric Function Theory: Explorations in Complex Analysis, Birkhäuser Publishing, Boston, 2006.
S. G. Krantz, Invariant metrics and the boundary behavior of holomorphic functions on domains in \({\mathbb{C}}^{n}\), Jour. Geometric. Anal. 1(1991), 71–98.
S. G. Krantz, Calculation and estimation of the Poisson kernel, J. Math. Anal. Appl. 302(2005)143–148.
S. G. Krantz, A new proof and a generalization of Ramadanov’s theorem, Complex Variables and Elliptic Eq. 51(2006), 1125–1128.
S. G. Krantz, Complex Analysis: The Geometric Viewpoint, 2nd ed., Mathematical Association of America, Washington, D.C., 2004.
S. G. Krantz, Canonical kernels versus constructible kernels, preprint.
S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Expositiones Math. 3(1983), 193–260.
S. G. Krantz, Characterizations of smooth domains in \(\mathbb{C}\) by their biholomorphic self maps, Am. Math. Monthly 90(1983), 555–557.
S. G. Krantz, A Guide to Functional Analysis, Mathematical Association of America, Washington, D.C., 2013, to appear.
S. G. Krantz, A direct connection between the Bergman and Szegő projections, Complex Analysis and Operator Theory, to appear.
S. G. Krantz and H. R. Parks, The Geometry of Domains in Space, Birkhäuser Publishing, Boston, MA, 1996.
S. G. Krantz and H. R. Parks, The Implicit Function Theorem, Birkhäuser, Boston, 2002.
S. G. Krantz and M. M. Peloso, The Bergman kernel and projection on non-smooth worm domains, Houston J. Math. 34 (2008), 9.3.-950.
S. G. Krantz and M. M. Peloso, Analysis and geometry on worm domains, J. Geom. Anal. 18(2008), 478–510.
L. Lempert, La metrique Kobayashi et las representation des domains sur la boule, Bull. Soc. Math. France 109(1981), 427–474.
S.-Y. Li, S-Y. Li, Neumann problems for complex Monge–Ampère equations, Indiana University J. of Math, 43(1994), 1099–1122.
E. Ligocka, The Sobolev spaces of harmonic functions, Studia Math. 54(1986). 79–87.
E. Ligocka, Remarks on the Bergman kernel function of a worm domain, Studia Mathematica 130(1998), 109–113.
B.-L. Min, Domains with prescribed automorphism group, J. Geom. Anal. 19 (2009), 911–928.
R. Narasimhan, Several Complex Variables, University of Chicago Press, Chicago, 1971.
L. Nirenberg, S. Webster, and P. Yang, Local boundary regularity of holomorphic mappings. Comm. Pure Appl. Math. 33(1980), 305–338.
T. Ohsawa, A remark on the completeness of the Bergman metric, Proc. Japan Acad. Ser. A Math. Sci., 57(1981), 238–240.
Painlevé, Sur les lignes singulières des functions analytiques, Thèse, Gauthier-Villars, Paris, 1887.
J. Peetre, The Berezin transform and Ha-Plitz operators, J. Operator Theory 24(1990), 165–186.
P. Petersen, Riemannian Geometry, Springer, New York, 2009.
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms, Acta Math. 157(1986), 99–157.
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. I. Acta Math. 157(1986), 99–157.
D. H. Phong and E. M. Stein, Hilbert integrals, singular integrals, and Radon transforms. II. Invent. Math. 86(1986), 75–113.
S. Pinchuk, The scaling method and holomorphic mappings, Several Complex Variables and Complex Ggeometry, Part 1 (Santa Cruz, CA, 1989), 151–161, Proc. Sympos. Pure Math., 52, Part 1, Amer. Math. Soc., Providence, RI, 1991.
S. Pinchuk and S. V. Hasanov, Asymptotically holomorphic functions (Russian), Mat. Sb. 134(176) (1987), 546–555.
S. Pinchuk and S. I. Tsyganov, Smoothness of CR-mappings between strictly pseudoconvex hypersurfaces. (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 53(1989), 1120–1129, 1136; translation in Math. USSR-Izv. 35(1990), 457–467.
I. Ramadanov, Sur une propriété de la fonction de Bergman. (French) C. R. Acad. Bulgare Sci. 20(1967), 759–762.
I. Ramadanov, A characterization of the balls in \({\mathbb{C}}^{n}\) by means of the Bergman kernel, C. R. Acad. Bulgare Sci. 34(1981), 927–929.
R. M. Range, A remark on bounded strictly plurisubharmonic exhaustion functions, Proc. A.M.S. 81(1981), 220–222.
S. Roman, The formula of Faà di Bruno, Am. Math. Monthly 87(1980), 805–809.
B. Rodin and S. Warschawski, Estimates of the Riemann mapping function near a boundary point, in Romanian-Finnish Seminar on Complex Analysis, Springer Lecture Notes, vol. 743, 1979, 349–366.
J.-P. Rosay, Sur une characterization de la boule parmi les domains de \({\mathbb{C}}^{n}\) par son groupe d’automorphismes, Ann. Inst. Four. Grenoble XXIX(1979), 91–97.
P. Rosenthal, On the zeroes of the Bergman function in doubly-connected domains, Proc. Amer. Math. Soc. 21(1969), 33–35.
W. Rudin, Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976.
W. Rudin, Function Theory in the Unit Ball of \({\mathbb{C}}^{n}\), Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Springer, Berlin, 1980.
S. Semmes, A generalization of Riemann mappings and geometric structures on a space of domains in \({\mathbb{C}}^{n},\) Memoirs of the American Mathematical Society, 1991.
N. Sibony, A class of hyperbolic manifolds, Ann. of Math. Stud. 100(1981), 357–372.
Y.-T. Siu, Non Hölder property of Bergman projection of smooth worm domain, Aspects of Mathematics—Algebra, Geometry, and Several Complex Variables, N. Mok (ed.), University of Hong Kong, 1996, 264–304.
M. Skwarczynski, The distance in the theory of pseudo-conformal transformations and the Lu Qi-King conjecture, Proc. A.M.S. 22(1969), 305–310.
E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ. 1970.
E. M. Stein, Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton University Press, Princeton, 1972.
E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Space, Princeton University Press, Princeton, NJ, 1971.
K. Stromberg, An Introduction to Classical Real Analysis, Wadsworth, Belmont, 1981.
N. Suita and A. Yamada On the Lu Qi-Keng conjecture, Proc. A.M.S. 59(1976), 222–224.
G. Szegő, Über Orthogonalsysteme von Polynomen, Math. Z. 4(1919), 139–151.
N. Tanaka, On generalized graded Lie algebras and geometric structures, I, J. Math. Soc. Japan 19(1967), 215–254.
G. B. Thomas, Calculus, 7th ed., Addison-Wesley, Reading, MA, 1999.
F. Treves, Introduction to Pseudodifferential and Fourier Integral Operators, Vol. II, Plenum Press, New York 1982.
S. Warschawski, On the boundary behavior of conformal maps, Nagoya Math. J. 30(1967), 83–101.
S. Warschawski, On boundary derivatives in conformal mapping, Ann. Acad. Sci. Fenn. Ser. A I no. 420(1968), 22 pp.
S. Warschawski, Hölder continuity at the boundary in conformal maps, J. Math. Mech. 18(1968/9), 423–7.
S. Webster, Biholomorphic mappings and the Bergman kernel off the diagonal, Invent. Math. 51(1979), 155–169.
S. Webster, On the reflection principle in several complex variables, Proc. Amer. Math. Soc. 71(1978), 26–28.
E. Whittaker and G. Watson, A Course of Modern Analysis, 4th ed., Cambridge Univ. Press, London, 1935.
J. Wiegerinck, Domains with finite dimensional Bergman space, Math. Z. 187(1984), 559–562.
B. Wong, Characterization of the ball in \({\mathbb{C}}^{n}\) by its automorphism group, Invent. Math. 41(1977), 253–257.
S.-T. Yau, Problem section, Seminar on Differential Geometry, S.-T. Yau ed., Annals of Math. Studies, vol. 102, Princeton University Press, 1982, 669–706.
K. Zhu, Spaces of Holomorphic Functions in the Unit Ball, Springer, New York, 2005.
Author information
Authors and Affiliations
APPENDIX: Scaling in Dimension One
APPENDIX: Scaling in Dimension One
7.1.1 The Scaling of the Unit Disc
Let D be the open unit disc in the complex plane \(\mathbb{C}\). Choose a sequence a j in D satisfying the conditions
and
Consider the sequence of dilations
Let us write \(\lambda _{j} = 1 - a_{j}\). Then one sees immediately that
It follows that the sequence of sets L j (D) converges to the left half plane \(H =\{\zeta \in \mathbb{C}\mid Re\,\zeta < 0\}\) in the sense that
and
[Compare the concept of convergence in the Hausdorff metric on sets—see [FED].]
Now we combine this simple observation with the fact that there exists the sequence of maps
that are automorphisms of D satisfying \(\varphi _{j}(0) = a_{j}\). Consider the sequence of composite maps
A direct computation yields that
Hence, in fact we see that the sequence of holomorphic mappings \(L_{j} \circ \varphi _{j}\) converges uniformly on compact subsets of D to the mapping
that is a biholomorphic mapping from the open unit disc D onto the left half plane H. (We have in effect discovered here a means to see the Cayley map by way of scaling.)
The point is that we have exploited the automorphism of the disc to see that the disc is conformally equivalent to a certain canonical domain—namely, the half plane. This result is neither surprising nor insightful. But it is a toy version of the main results that we shall present below.
7.1.2 A Generalization
We now expand the simple observations of the preceding subsection to yield the statement and the proof of the following one-dimensional version of the Wong–Rosay theorem:
Proposition:
Let Ω be a domain in the complex plane \(\mathbb{C}\) admitting a boundary point p such that
-
There exists an open neighborhood U of p in \(\mathbb{C}\) such that \(U \cap \partial \Omega \) is a C 1 curve.
-
There exists a sequence \(\varphi _{j}\) of automorphisms of Ω and a point q ∈ Ω such that
$$\displaystyle{\lim _{j\rightarrow \infty }\varphi _{j}(q) = p.}$$
Then Ω is biholomorphic to the open unit disc.
See [KRA13] for this theorem. We use this simple result to illustrate the technique of scaling.
In order to be consistent with the remainder of this chapter, we change a bit the notation for the orbit accumulation point and the point whose orbit we are calculating. This will all make sense in context.
Sketch of the Proof: Let \(q_{j} =\varphi _{j}(q)\) for each j. Choose the closest point in the boundary to q j and call it p j . If the closest boundary point p j to q j is not unique, then make a choice. As j tends to infinity, p j converges to p because q j converges to p. Then we select θ j and apply the map \(\rho _{j}(z) \equiv {\mathrm{e}}^{i\theta _{j}}(z - p_{j})\) so that
for each j. Now consider the sequence of mappings
Notice that \(\psi _{j}(\Omega ) = \frac{1} {\rho (q_{j})}\ \rho _{j}(\Omega )\) for each j. Thus we expect that ψ j (Ω) is almost the right half plane as j becomes very large. At least every ψ j (Ω) is contained in \(\mathbb{C}\setminus \ell\) for some line segment ℓ of positive length and for every j. (Note that ℓ can be chosen independently of j.) Therefore one can select a subsequence from \(\{\psi _{j}\}\) that converges uniformly on compact subsets of Ω. Let \(\hat{\psi }\) be the limit mapping. Then we expect \(\hat{\psi }: \Omega \rightarrow \mathbb{C}\) to be an injective holomorphic mapping, and furthermore, \(\hat{\psi }(\Omega )\) is equal to the right half plane. Thus we hope to conclude that Ω is biholomorphic to the right half plane, which in turn is biholomorphic to the open unit disc. See Fig. 7.1.
This plan actually works, but it is evident that there are several points that need clarification. We shall now present the precise proof, which will show much of the essence of the scaling method.
Rigorous Proof of the Main Result: Keeping the “Plan of the Proof” in mind, we present the precise proof in several steps. Let p ∈ ∂ Ω be as in the hypothesis of the proposition. Write \(D(p,r) =\{ z \in \mathbb{C}\mid \vert z - p\vert < r\}\). Transforming Ω by a conformal mapping z↦eiα(z − p), we may assume the following with no loss of generality:
-
(a)
p = 0
-
(b)
\(\Omega \cap D(p,r) =\{ z = x + \mathit{iy}\mid y >\psi (x),\vert z - p\vert < r\}\) and \(\partial \Omega \cap D(p,r) =\{ z\mid y =\psi (x),\vert z - p\vert < r\}\) for a real-valued C 1 function ψ in one real variable satisfying ψ(0) = 0 and ψ ′(0) = 0.
Step 1. The Scaling Map. Notice that the sequence \(\varphi _{j}(q)\) now converges to 0 as j → ∞. For each j, we choose a point p j ∈ ∂ Ω that is the closest to \(\varphi _{j}(q)\). Since p j also converges to 0, replacing \(\varphi _{j}\) by a subsequence if necessary, we may assume that every p j ∈ D(p, r ∕ 4). Now, for each j, set
Notice that \(\varphi _{j}(q) - p_{j}\) is a positive scalar multiple of the inward unit normal vector to ∂ Ω at p j . Thus \(\frac{\varphi _{j}(q) - p_{j}} {\vert \varphi _{j}(q) - p_{j}\vert }\) converges to the inward unit normal vector to ∂ Ω at 0. This implies that α j in fact converges to the identity map. Consequently, there exist positive constants r 1, r 2 independent of j such that, for each j, there exists a C 1 function ψ j (x) defined for | x | < r 1 satisfying
Furthermore, for each ε > 0, there exists δ > 0 such that
regardless of j.
Next, let \(\lambda _{j} = \vert \varphi _{j}(q) - p_{j}\vert \) for each j. Consider the dilation map
Then the sequence of holomorphic mappings we want to construct is given by
Before starting the next step, we make a few remarks. The automorphism \(\varphi _{j}\) preserves the domain Ω but moves q to \(\varphi _{j}(q)\) so that \(\varphi _{j}(q)\) converges to the origin—recall that we made changes so that p became the origin at the beginning of the proof. Then the affine map α j adjusts Ω so that the direction vector \(\frac{\varphi _{j}(q) - p_{j}} {\vert \varphi _{j}(q) - p_{j}\vert }\) is transformed to a purely imaginary number. The final component L j in the construction simply magnifies the domain α j (Ω), while the map L j itself diverges.
Step 2. Convergence of the ψ j . We shall actually choose a subsequence from {ψ j } that converges uniformly on compact subsets of Ω. Observe first that
since \(\varphi _{j}(\Omega ) = \Omega \). Choosing a subsequence of ψ j , we may assume that λ j < 1 for every j. Then, since L j is a simple dilation by a positive number, and since α j (Ω) will miss a line segment
for some constant b independent of j, we see immediately that
for every j = 1, 2, …. Therefore Montel’s theorem implies that every subsequence of {ψ j } admits a subsequence, which we again (by an abuse of notation) denote by ψ j , that converges uniformly on compact subsets of Ω. Denote by \(\hat{\psi }\) the limit of the sequence ψ j .
Step 3. Analysis of \(\hat{\psi }(\Omega )\). We want to establish that
where \(\mathcal{U}\equiv \{ z \in \mathbb{C}\mid \ Im\,z > 0\}\).
Let ε be a positive real number and let K an arbitrary compact subset of Ω. We will show that \(\hat{\psi }(K) \subset C_{\epsilon }\), where \(C_{\epsilon } \equiv \{ z \in \mathbb{C}: -\epsilon <\arg z <\pi +\epsilon \}\).
Choose R > 0 such that \(\hat{\psi }(K)\) is contained in the disc D(0, R) of radius R centered at 0.
The sequence \(\varphi _{j}: \Omega \rightarrow \Omega \) is a normal family since \(\mathbb{C} \setminus \Omega \) contains a line segment with positive length. Every subsequence of \(\varphi _{j}\) contains a subsequence that converges uniformly on compact subsets, since \(\varphi _{j}(q)\) converges to p. Let \(g: \Omega \rightarrow \overline{\Omega }\) be a subsequential limit map. Then g(q) = p. Recall that \(p \in \partial \Omega \). Hence, the open mapping theorem yields that g(z) = p for every z ∈ Ω. Thus the sequence \(\varphi _{j}\) itself converges uniformly on compact subsets to the constant map with value p. Therefore we may choose N > 0 such that \(\varphi _{j}(K)\) is contained in a sufficiently small neighborhood of the origin for every j > N, and hence \(\alpha _{j} \circ \varphi _{j}(K) \subset C_{\epsilon }\) for every j > N. Then it follows immediately that \(\psi _{j}(K) \subset C_{\epsilon }\) for every j > N and consequently that
Since K is an arbitrary compact subset of Ω, it follows that \(\hat{\psi }(\Omega ) \subset \overline{\mathcal{U}}\). We also have \(\hat{\psi }(q) = i\), since \(\psi _{j}(q) = L_{j} \circ \alpha _{j} \circ \varphi _{j}(q) = i\) for every j = 1, 2, …. Therefore \(\hat{\psi }(\Omega ) \subset \mathcal{U}\).
Step 4. Convergence of \(\psi _{j}^{-1}\). Let \(\tilde{K}\) be an arbitrary compact subset of the upper half plane \(\mathcal{U}\). Then choose ε > 0 so that \(\tilde{K} \subset C_{\epsilon }\). Choose then r > 0 such that
Shrinking r > 0 if necessary, since α j converges to the identity map uniformly on compact subsets of \(\mathbb{C}\), there exists N > 0 such that
for every j > N. Hence, we see that \(\psi _{j}^{-1}\) maps K into Ω. Since \(\Omega \subset \mathbb{C} \setminus E\) as observed before, we may again choose a subsequence of ψ j , which we again denote by ψ j , so that \(\psi _{j}^{-1}\) converges to a holomorphic map, say \(\tau: \mathcal{U}\rightarrow \overline{\Omega }\). Since τ is holomorphic and τ(i) = q, we see that τ maps the upper half plane \(\mathcal{U}\) into Ω.
Step 5. Synthesis. We are ready to complete the proof. By the Cauchy estimates, the derivatives dψ j of ψ j as well as the derivatives \(d[\psi _{j}^{-1}]\) both converge. Therefore \(d\hat{\psi }(q) \cdot d\hat{\tau }(i) = 1\). This means that \(\hat{\psi }\circ \tau: \mathcal{U}\rightarrow \mathcal{U}\) is a holomorphic mapping satisfying \(\hat{\psi }\circ \tau (i) = i\) and \((\hat{\psi }\circ \tau )^{\prime}(i) = 1\). Then, by the Schwarz’s lemma, one concludes that \(\hat{\psi }\circ \tau = \mbox{ id}\), where id is the identity mapping. Likewise, the same reasoning applied to \(\tau \circ \hat{\psi }: \Omega \rightarrow \Omega \) implies that \(\tau \circ \hat{\psi } = \mbox{ id}\). So \(\hat{\psi }: \Omega \rightarrow \mathcal{U}\) is a biholomorphic mapping. □
Remark: The sequence of mappings ψ j constructed above is often called a scaling sequence. It is constructed from a composition of
-
The automorphisms carrying one fixed interior point successively to a boundary point
-
Certain affine adjustments
-
The stretching dilation map
The proof given above is a good example of the scaling technique. The main thrust of the method is that the image of the limit mapping is determined solely by the affine adjustments and the dilations, while the scaling sequence converges to a conformal mapping. □
Remark: As observed earlier, the main result here can be proved in a much simpler way. Namely, one may conclude immediately from the argument on the shrinking of \(\varphi _{j}(K)\) into a simply connected subset of Ω that Ω must be simply connected. Then the conclusion follows by the Riemann mapping theorem. But we are trying to skirt around the Riemann mapping theorem. The goal of this argument is to provide a basis for the scaling method which can be applied to the higher-dimensional cases. □
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Krantz, S.G. (2013). Curvature of the Bergman Metric. In: Geometric Analysis of the Bergman Kernel and Metric. Graduate Texts in Mathematics, vol 268. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-7924-6_7
Download citation
DOI: https://doi.org/10.1007/978-1-4614-7924-6_7
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-7923-9
Online ISBN: 978-1-4614-7924-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)