Complex scaling and geometric analysis of several variables

다국어 초록 (Multilingual Abstract)

The purpose of this paper is to
survey the use of the important method of scaling
in analysis, and particularly in complex analysis.
Applications are given to the study of automorophism
groups, to canonical kernels, to holomorphic invariants, and
to analysis in infinite dimensions. Current research directions
are described and future paths indicated.

다국어 초록 (Multilingual Abstract)

The purpose of this paper is to
survey the use of the important method of scaling
in analysis, and particularly in complex analysis.
Applications are given to the study of automorophism
groups, to canonical kernels, to holomorphic invariants, and
to analysis in infinite dimensions. Current research directions
are described and future paths indicated.

참고문헌 (Reference)

1 J. Byun, "Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group" 12 (12): 581-599, 2002

2 P. Duren, "Univalent Functions" Springer-Verlag 1983

3 M. Landucci, "Unbounded domains in C2 with non-compact automorphisms group" 42 (42): 300-307, 2002

4 S. Kobayashi, "Transformation Groups in Differential Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70" Springer-Verlag 1972

5 S. Pinchuk, "The scaling method and holomorphic mappings" Amer. Math. Soc. 52 : 151-161, 1991

6 R. Saerens, "The isometry groups of manifolds and the automorphism groups of domains" 301 (301): 413-429, 1987

7 J. Moser, "The holomorphic equivalence of real hypersurfaces" 659-668, 1980

8 G. Aladro, "The comparability of the Kobayashi approach region and the admissible approach region" 33 (33): 42-63, 1989

9 R. E. Greene, "The automorphism groups of strongly pseudoconvex domains" 261 (261): 425-446, 1982

10 M. Landucci, "The automorphism group of domains with boundary points of infinite type" 48 (48): 875-885, 2004

11 C. Fefferman, "The Bergman kernel and biholomorphic mappings of pseudoconvex domains" 26 : 1-65, 1974

12 J. P. Rosay, "Sur une caract´erisation de la boule parmi les domaines de Cn par son groupe d’automorphismes" 29 (29): 91-97, 1979

13 J. McNeal, "Subelliptic estimates and scaling in the ∂-Neumann problem, Explorations in complex and Riemannian geometry" Amer. Math. Soc. 332 : 197-217, 2003

14 J. P. D’Angelo, "Subelliptic estimates and finite type, Several complex variables (Berkeley, CA, 1995–1996)" Cambridge Univ. Press 199-232, 1999

15 K. H. Lee, "Strongly pseudoconvex domains in almost complex manifolds"

16 M. Kuranishi, "Strongly pseudoconvex CR structures over small balls. III. An embedding theorem" 116 (116): 249-330, 1982

17 D. Ma, "Sharp estimates of the Kobayashi metric near strongly pseudoconvex points, The Madison Symposium on Complex Analysis (Madison, WI, 1991)" 137 : 329-338, 1992

18 J. P. D’Angelo, "Several Complex Variables and the Geometry of Real Hypersurfaces" CRC Press 1993

19 R. Narasimhan, "Several Complex Variables" University of Chicago Press 1971

20 X. Huang, "Schwarz reflection principle in complex spaces of dimension two" 21 (21): 1787-1828, 1996

21 M. Christ, "Regularity properties of the ∂b equation on weakly pseudoconvex CR manifolds of dimension 3" 1 (1): 587-646, 1988

22 K. Diederich, "Reflection principle in higher dimensions" 703-712, 1998

23 J. Winkelmann, "Realizing connected Lie groups as automorphism groups of complex manifolds" 79 (79): 285-299, 2004

24 S. S. Chern, "Real hypersurfaces in complex manifolds" 133 : 219-271, 1974

25 M. Gromov, "Pseudoholomorphic curves in symplectic manifolds" 82 (82): 307-347, 1985

26 K. Diederich, "Pseudoconvex domains with real-analytic boundary" 107 (107): 371-384, 1978

27 S. G. Krantz, "Partial Differential Equations and Complex Analysis" CRC Press 1992

28 S.Webster, "On the proof of Kuranishi’s embedding theorem" 6 (6): 183-207, 1989

29 H. Gaussier, "On the geometry of model almost complex manifolds with boundary" 254 (254): 567-589, 2006

30 V. Ejov, "On the dimension of the stability group for a Levi non-degenerate hypersurface" 49 (49): 1155-1169, 2005

31 R. Schoen, "On the conformal and CR automorphism groups" 5 (5): 464-481, 1995

32 J. Byun, "On the compactness of the automorphism group of a domain" 341 (341): 545-548, 2005

33 J. Byun, "On the boundary accumulation points for the holomorphic automorphism groups" 51 (51): 379-386, 2003

34 K.-T. Kim, "On the automorphism groups of convex domains in Cn" 4 (4): 33-40, 2004

35 J. Byun, "On the automorphism group of the Kohn-Nirenberg domain" 266 (266): 342-356, 2002

36 S.Webster, "On the Moser normal form at a non-umbilic point" 233 (233): 97-102, 1978

37 N. Tanaka, "On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections" 2 (2): 131-190, 1976

38 H.Wu, "Old and new invariant metrics on complex manifolds" 38 : 640-682, 1993

39 J. Moser, "Normal forms for real surfaces in C2 near complex tangents and hyperbolic surface transformations" 150 (150): 255-296, 1983

40 K.-T. Kim, "Normal analytic polyhedra in C2 with a noncompact automorphism group" 11 (11): 283-293, 2001

41 J. McNeal, "Local geometry of decoupled pseudoconvex domains" 223-230, 1991

42 S. Sternberg, "Local contractions and a theorem of Poincar´e" 79 : 809-824, 1957

43 N. Kruzhilin, "Linearization of local automorphisms of pseudoconvex surf" 271 (271): 280-282, 1983

44 V. Ezhov, "Linearization of automorphisms of a real-analytic hypersurface" 49 (49): 731-765, 1985

45 L. Nirenberg, "Lectures on linear partial differential equations" Journal of the American Mathematical Society 1973

46 L. H"Normander, "L2 estimates and existence theorems for the "P∂ operator" 113 : 89-152, 1965

47 P. Klembeck, "K¨ahler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets" 27 (27): 275-282, 1978

48 R. E. Greene, "Invariants of Bergman geometry and the automorphism groups of domains in Cn, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989)" EditEl 107-136, 1991

49 C. Fefferman, "H¨older estimates on domains of complex dimension two and on three-dimensional CR manifolds" 69 (69): 223-303, 1988

50 S. Kobayashi, "Hyperbolic Manifolds and Holomorphic Mappings" Marcel Dekker 1970

51 J. Moser, "Holomorphic equivalence and normal forms of hypersurfaces" 109-112, 1975

52 X. Ma, "Holomorphic Morse Inequalities and Bergman Kernels" 2007

53 L. K. Hua, "Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains" American Mathematical Society 1963

54 R. E. Greene, "Geometric foundations for analysis on complex domains" 1995

55 X. Ma, "Generalized Bergman kernels on symplectic manifolds" 339 (339): 439-498, 2004

56 S. G. Krantz, "Function Theory of Several Complex Variables" American Mathematical Society 2000

57 J. McNeal, "Estimates on the Bergman kernels of convex domains" 109 (109): 108-139, 1994

58 H. Gaussier, "Estimates of the Kobayashi-Royden metric in almost complex manifolds" 133 (133): 259-273, 2005

59 D. Catlin, "Estimates of invariant metrics on pseudoconvex domains of dimension two" 200 (200): 429-466, 1989

60 A. Nagel, "Estimates for the Bergman and Szeg¨o kernels in C2" 129 (129): 113-149, 1989

61 A. V. Isaev, "Effective actions of the unitary group on complex manifolds" 54 (54): 1254-1279, 2002

62 K.-T. Kim, "Dynamics of local automorphisms of embedded CRmanifolds" 76 (76): 477-480, 2004

63 A Isaev, "Domains with non-compact automorphism group: a survey" 146 (146): 1-38, 1999

64 K. H. Lee, "Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point" 54 (54): 179-205, 2006

65 E. Bedford, "Domains in Cn+1 with noncompact automorphism group" 1 (1): 165-191, 1991

66 K.-T. Kim, "Domains in Cn with a piecewise Levi flat boundary which possess a noncompact automorphism group" 292 (292): 575-586, 1992

67 E. Bedford, "Domains in C2 with noncompact automorphism groups" 47 (47): 199-222, 1998

68 S. Helgason, "Differential Geometry and Symmetric Spaces" Academic Press 1962

69 D. Burns, "Deformations of strictly pseudoconvex domains" 46 (46): 237-253, 1978

70 R. E. Greene, "Deformation of complex structures, estimates for the "P∂ equation, and stability of the Bergman kernel" 43 (43): 1-86, 1982

71 K.-T. Kim, "Complex scaling and domains with non-compact automorphism group" 45 (45): 1273-1299, 2001

72 S. Frankel, "Complex geometry of convex domains that cover varieties" 163 (163): 109-149, 1989

73 A. Huckleberry, "Classification Theorems for Almost Homogeneous Spaces" Universit"Le de Nancy, Institut Elie Cartan 1984

74 R. E. Greene, "Characterizations of certain weakly pseudoconvex domains with noncompact automorphism groups, Complex analysis (University Park, Pa., 1986)" Springer 121-157, 1987

75 B. Wong, "Characterization of the unit ball in Cn by its automorphism group" 41 (41): 253-257, 1977

76 K.-T. Kim, "Characterization of the Hilbert ball by its automorphism group" 354 (354): 2797-2818, 2002

77 F. Berteloot, "Characterization of models in C2 by their automorphism groups" 5 (5): 619-634, 1994

78 S. G. Krantz, "Calculation and estimation of the Poisson kernel" 302 (302): 143-148, 2005

79 K.-T. Kim, "CR hypersurfaces with a weakly-contracting automorphism"

80 A. Bogges, "CR Manifolds and the Tangential Cauchy-Riemann Complex" CRC Press 1991

81 E. Bedford, "Bounded domains with prescribed group of automorphisms" 62 (62): 561-572, 1987

82 H. Boas, "Boundary limits of the Bergman kernel and metric" 42 (42): 449-461, 1995

83 J. J. Kohn, "Boundary behavior of δ on weakly pseudo-convex manifolds of dimension two" 6 : 523-542, 1972

84 I. Graham, "Boundary behavior of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in Cn with smooth boundary" 207 : 219-240, 1975

85 J. McNeal, "Boundary behavior of the Bergman kernel function in C2" 58 (58): 499-512, 1989

86 K.-T. Kim, "Boundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domains" 176 (176): 141-163, 1996

87 R. E. Greene, "Biholomorphic self-maps of domains, Complex analysis, II (College Park, Md., 1985–6)" Springer 136-207, 1987

88 S. R. Bell, "Biholomorphic mappings and the "P∂-problem" 114 (114): 103-113, 1981

89 K. H. Lee, "Automorphism groups of almost complex manifolds" Pohang University of Science and Technology (POSTECH) 2005

90 S. Fu, "Asymptotic expansions of invariant metrics of strictly pseudoconvex domains" 38 (38): 196-206, 1995

91 K.-T. Kim, "Asymptotic behavior of the curvature of the Bergman metric of the thin domains" 155 (155): 99-110, 1992

92 S. Lee, "Asymptotic behavior of the Kobayashi metric on certain infinite-type pseudoconvex domains in C2" 256 (256): 190-215, 2001

93 K.-T. Kim, "Asymptotic behavior of the Bergman kernel and associated invariants in certain infinite type pseudoconvex domains" 14 (14): 775-795, 2002

94 K.-T. Kim, "Analytic polyhedra in C2 with a non-compact automorphism group" 579 : 1-12, 2005

95 K. H. Lee, "Almost complex manifolds and Cartan’s uniqueness theorem" 358 (358): 2057-2069, 2006

96 S. Bell, "A simplification and extension of Fefferman’s theorem on biholomorphic mappings" 57 (57): 283-289, 1980

97 J. J. Kohn, "A pseudo-convex domain not admitting a holomorphic support function" 201 : 265-268, 1973

98 K.-T. Kim, "A note on: “Characterization of the Hilbert ball by its automorphisms” [J. Korean Math. Soc. 40 (2003), no. 3, 503–16; MR1973915]" 309 (309): 761-763, 2005

99 T. Akahori, "A new approach to the local embedding theorem of CR-structures for n ≥ 4 (the local solvability for the operator ∂b in the abstract sense)" 67 (67): 1987

100 T. Bloom, "A geometric characterization of points of type m on real submanifolds of Cn" 12 (12): 171-182, 1977

101 J. P. D’Angelo, "A gentle introduction to points of finite type on real hypersurfaces, Explorations in complex and Riemannian geometry" 332 : 19-26, 2003