JOURNAL BROWSE
Search
Advanced SearchSearch Tips
COMPLEX SCALING AND GEOMETRIC ANALYSIS OF SEVERAL VARIABLES
facebook(new window)  Pirnt(new window) E-mail(new window) Excel Download
 Title & Authors
COMPLEX SCALING AND GEOMETRIC ANALYSIS OF SEVERAL VARIABLES
Kim, Kang-Tae; Krantz, Steven G.;
  PDF(new window)
 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.
 Keywords
automorphism group;scaling;pseudo convexity;finite type;isotropy group;orbit;domain;
 Language
English
 Cited by
1.
Complete prolongation for infinitesimal automorphisms on almost complex manifolds, Mathematische Zeitschrift, 2010, 264, 4, 913  crossref(new windwow)
2.
Gromov hyperbolicity and the Kobayashi metric on convex domains of finite type, Mathematische Annalen, 2016, 365, 3-4, 1425  crossref(new windwow)
3.
Integrable Submanifolds in Almost Complex Manifolds, Journal of Geometric Analysis, 2010, 20, 1, 177  crossref(new windwow)
4.
The automorphism groups of domains in complex space: a survey, Quaestiones Mathematicae, 2013, 36, 2, 225  crossref(new windwow)
 References
1.
T. Akahori, A new approach to the local embedding theorem of CR-structures for n $\geq$ 4 (the local solvability for the operator $ \partial$b in the abstract sense), Mem. Amer. Math. Soc. 67 (1987), no. 366, xvi+257 pp

2.
G. Aladro, The comparability of the Kobayashi approach region and the admissible approach region, Illinois J. Math. 33 (1989), no. 1, 42-63

3.
E. Bedford and J. Dadok, Bounded domains with prescribed group of automorphisms, Comment. Math. Helv. 62 (1987), no. 4, 561-572 crossref(new window)

4.
E. Bedford and S. Pinchuk, Domains in $C^n+1$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), no. 3, 165-191 crossref(new window)

5.
E. Bedford and S. Pinchuk, Domains in $C^2$ with noncompact automorphism groups, Indiana Univ. Math. J. 47 (1998), no. 1, 199-222

6.
S. R. Bell, Biholomorphic mappings and the $\partial$-problem, Ann. of Math. (2) 114 (1981), no. 1, 103-113 crossref(new window)

7.
S. Bell and E. Ligocka, A simplification and extension of Fefferman's theorem on biholomorphic mappings, Invent. Math. 57 (1980), no. 3, 283-289 crossref(new window)

8.
F. Berteloot, Characterization of models in $C^2$ by their automorphism groups, Internat. J. Math. 5 (1994), no. 5, 619-634 crossref(new window)

9.
T. Bloom and I. Graham, A geometric characterization of points of type m on real submanifolds of $C^n$, J. Differential Geometry 12 (1977), no. 2, 171-182.

10.
H. Boas, E. Straube, and J. Yu, Boundary limits of the Bergman kernel and metric, Michigan Math. J. 42 (1995), no. 3, 449-461 crossref(new window)

11.
A. Bogges, CR Manifolds and the Tangential Cauchy-Riemann Complex, Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1991

12.
D. Burns, S. Shnider, and R. O. Wells, Deformations of strictly pseudoconvex domains, Invent. Math. 46 (1978), no. 3, 237-253 crossref(new window)

13.
J. Byun, On the automorphism group of the Kohn-Nirenberg domain, J. Math. Anal. Appl. 266 (2002), no. 2, 342-356 crossref(new window)

14.
J. Byun, On the boundary accumulation points for the holomorphic automorphism groups, Michigan Math. J. 51 (2003), no. 2, 379-386 crossref(new window)

15.
J. Byun and H. Gaussier, On the compactness of the automorphism group of a domain, C. R. Math. Acad. Sci. Paris 341 (2005), no. 9, 545-548 crossref(new window)

16.
J. Byun, H. Gaussier, and K.-T. Kim, Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group, J. Geom. Anal. 12 (2002), no. 4, 581-599 crossref(new window)

17.
D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), no. 3, 429-466 crossref(new window)

18.
S. S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271 crossref(new window)

19.
M. Christ, Regularity properties of the $\partial$b equation on weakly pseudoconvex CR manifolds of dimension 3, J. Amer. Math. Soc. 1 (1988), no. 3, 587-646 crossref(new window)

20.
J. P. D'Angelo, Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, FL, 1993

21.
J. P. D'Angelo, A gentle introduction to points of finite type on real hypersurfaces, Explorations in complex and Riemannian geometry, 19-36, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003 crossref(new window)

22.
J. P. D'Angelo and J. J. Kohn, Subelliptic estimates and finite type, Several complex variables (Berkeley, CA, 1995-1996), 199-232, Math. Sci. Res. Inst. Publ., 37, Cambridge Univ. Press, Cambridge, 1999

23.
K. Diederich and J. E. Fornaess, Pseudoconvex domains with real-analytic boundary, Ann. Math. (2) 107 (1978), no. 2, 371-384 crossref(new window)

24.
K. Diederich and S. Pinchuk, Reflection principle in higher dimensions, Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998), Doc. Math. 1998, Extra Vol. II, 703-712

25.
P. Duren, Univalent Functions, Grundlehren der Mathematischen Wissenschaften, 259. Springer-Verlag, New York, 1983

26.
V. Ejov and A. Isaev, On the dimension of the stability group for a Levi non-degenerate hypersurface, Illinois J. Math. 49 (2005), no. 4, 1155-1169

27.
V. Ezhov, Linearization of automorphisms of a real-analytic hypersurface, Izv. Akad. Nauk SSSR Ser. Mat. 49 (1985), no. 4, 731-765

28.
C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1-65 crossref(new window)

29.
C. Fefferman and J. Kohn, Holder estimates on domains of complex dimension two and on three-dimensional CR manifolds, Adv. in Math. 69 (1988), no. 2, 223-303 crossref(new window)

30.
S. Frankel, Complex geometry of convex domains that cover varieties, Acta Math. 163 (1989), no. 1-2, 109-149 crossref(new window)

31.
S. Fu, Asymptotic expansions of invariant metrics of strictly pseudoconvex domains, Canad. Math. Bull. 38 (1995), no. 2, 196-206 crossref(new window)

32.
H. Gaussier and A. Sukhov, On the geometry of model almost complex manifolds with boundary, Math. Z. 254 (2006), no. 3, 567-589 crossref(new window)

33.
H. Gaussier and A. Sukhov, Estimates of the Kobayashi-Royden metric in almost complex manifolds, Bull. Soc. Math. France 133 (2005), no. 2, 259-273

34.
I. Graham, Boundary behavior of the Caratheodory and Kobayashi metrics on strongly pseudoconvex domains in Cn with smooth boundary, Trans. Amer. Math. Soc. 207 (1975), 219-240 crossref(new window)

35.
R. E. Greene and S. 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 crossref(new window)

36.
R. E. Greene and S. G. Krantz, Characterizations of certain weakly pseudoconvex domains with noncompact automorphism groups, Complex analysis (University Park, Pa., 1986), 121-157, Lecture Notes in Math., 1268, Springer, Berlin, 1987

37.
R. E. Greene and S. G. Krantz, Biholomorphic self-maps of domains, Complex analysis, II (College Park, Md., 1985-86), 136-207, Lecture Notes in Math., 1276, Springer, Berlin, 1987

38.
R. E. Greene and S. G. Krantz, The automorphism groups of strongly pseudoconvex domains, Math. Ann. 261 (1982), no. 4, 425-446 crossref(new window)

39.
R. E. Greene and S. G. Krantz, Invariants of Bergman geometry and the automorphism groups of domains in $C^n$, Geometrical and algebraical aspects in several complex variables (Cetraro, 1989), 107-136, Sem. Conf., 8, EditEl, Rende, 1991

40.
R. E. Greene and S. G. Krantz, Geometric foundations for analysis on complex domains, Proc. of the 1994 Conference in Cetraro (D. Struppa, ed.), 1995

41.
M. Gromov, Pseudoholomorphic curves in symplectic manifolds, Invent. Math. 82 (1985), no. 2, 307-347 crossref(new window)

42.
S. Helgason, Differential Geometry and Symmetric Spaces, Academic Press, New York, 1962

43.
L. Hormander, $L^2$ estimates and existence theorems for the $\partial$ operator, Acta Math. 113 (1965), 89-152 crossref(new window)

44.
L. K. Hua, Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains, American Mathematical Society, Providence, 1963

45.
X. Huang, Schwarz reflection principle in complex spaces of dimension two, Comm. Partial Differential Equations 21 (1996), no. 11-12, 1781-1828 crossref(new window)

46.
A. Huckleberry and E. Oeljeklaus, Classification Theorems for Almost Homogeneous Spaces, Institut Elie Cartan, 9. Universite de Nancy, Institut Elie Cartan, Nancy, 1984

47.
A Isaev and S. G. Krantz, Domains with non-compact automorphism group: a survey, Adv. Math. 146 (1999), no. 1, 1-38 crossref(new window)

48.
A. V. Isaev and N. G. Kruzhilin, Effective actions of the unitary group on complex manifolds, Canad. J. Math. 54 (2002), no. 6, 1254-1279 crossref(new window)

49.
K.-T. Kim, Domains in Cn with a piecewise Levi flat boundary which possess a noncompact automorphism group, Math. Ann. 292 (1992), no. 4, 575-586 crossref(new window)

50.
K.-T. Kim, On the automorphism groups of convex domains in Cn, Adv. Geom. 4 (2004), no. 1, 33-40 crossref(new window)

51.
K.-T. Kim, Asymptotic behavior of the curvature of the Bergman metric of the thin domains, Pacific J. Math. 155 (1992), no. 1, 99-110 crossref(new window)

52.
K.-T. Kim and S.-Y. Kim, CR hypersurfaces with a weakly-contracting automorphism, J. Geom. Anal. (To appear)

53.
K.-T. Kim and S. G. Krantz, Complex scaling and domains with non-compact automorphism group, Illinois J. Math. 45 (2001), no. 4, 1273-1299

54.
K.-T. Kim and S. G. Krantz, Characterization of the Hilbert ball by its automorphism group, Trans. Amer. Math. Soc. 354 (2002), no. 7, 2797-2818 crossref(new window)

55.
K.-T. Kim, S. G. Krantz, and A. Spiro, Analytic polyhedra in $C^2$ with a non-compact automorphism group, J. Reine Angew. Math. 579 (2005), 1-12

56.
K.-T. Kim and S. Lee, Asymptotic behavior of the Bergman kernel and associated invariants in certain infinite type pseudoconvex domains, Forum Math. 14 (2002), no. 5, 775-795 crossref(new window)

57.
K.-T. Kim and D. Ma, A note on: 'Characterization of the Hilbert ball by its automorphisms' J. Korean Math. Soc. 40 (2003), no. 3, 503-516

58.
K.-T. Kim and D. Ma, A note on: 'Characterization of the Hilbert ball by its automorphisms' MR1973915, J. Math. Anal. Appl. 309 (2005), no. 2, 761-763 crossref(new window)

59.
K.-T. Kim and A. Pagano, Normal analytic polyhedra in $C^2$ with a noncompact automorphism group, J. Geom. Anal. 11 (2001), no. 2, 283-293 crossref(new window)

60.
K.-T. Kim and G. Schmalz, Dynamics of local automorphisms of embedded CRmanifolds, Mat. Zametki 76 (2004), no. 3, 477-480 crossref(new window)

61.
K.-T. Kim and G. Schmalz, Dynamics of local automorphisms of embedded CRmanifolds, Mtranslation in Math. Notes 76 (2004), no. 3-4, 443-446 crossref(new window)

62.
K.-T. Kim and J. Yu, Boundary behavior of the Bergman curvature in strictly pseudoconvex polyhedral domains, Pacific J. Math. 176 (1996), no. 1, 141-163

63.
P. Klembeck, Kahler metrics of negative curvature, the Bergmann metric near the boundary, and the Kobayashi metric on smooth bounded strictly pseudoconvex sets, Indiana Univ. Math. J. 27 (1978), no. 2, 275-282 crossref(new window)

64.
S. Kobayashi, Hyperbolic Manifolds and Holomorphic Mappings, Marcel Dekker, New York, 1970

65.
S. Kobayashi, Transformation Groups in Differential Geometry, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 70, Springer-Verlag, New York-Heidelberg, 1972

66.
J. J. Kohn, Boundary behavior of $\delta$ on weakly pseudo-convex manifolds of dimension two, J. Differential Geometry 6 (1972), 523-542

67.
J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann. 201 (1973), 265-268 crossref(new window)

68.
S. G. Krantz, Function Theory of Several Complex Variables, American Mathematical Society, Providence, RI, 2000

69.
S. G. Krantz, Calculation and estimation of the Poisson kernel, J. Math. Anal. Appl. 302 (2005), no. 1, 143-148 crossref(new window)

70.
S. G. Krantz, Partial Differential Equations and Complex Analysis, CRC Press, Boca Raton, FL, 1992

71.
N. Kruzhilin and A. V. Loboda, Linearization of local automorphisms of pseudoconvex surfaces, Dokl. Akad. Nauk SSSR 271 (1983), no. 2, 280-282

72.
M. Kuranishi, Strongly pseudoconvex CR structures over small balls. III. An embedding theorem, Ann. of Math. (2) 116 (1982), no. 2, 249-330 crossref(new window)

73.
M. Landucci, The automorphism group of domains with boundary points of infinite type, Illinois J. Math. 48 (2004), no. 3, 875-885

74.
M. Landucci and G. Patrizio, Unbounded domains in $C^2$ with non-compact automorphisms group, Results Math. 42 (2002), no. 3-4, 300-307 crossref(new window)

75.
K. H. Lee, Automorphism groups of almost complex manifolds, Ph. D. dissertation, Pohang University of Science and Technology (POSTECH), Pohang 790-784 Korea, (2005), 97 pages

76.
K. H. Lee, Almost complex manifolds and Cartan's uniqueness theorem, Trans. Amer. Math. Soc. 358 (2006), no. 5, 2057-2069 crossref(new window)

77.
K. H. Lee, Domains in almost complex manifolds with an automorphism orbit accumulating at a strongly pseudoconvex boundary point, Michigan Math. J. 54 (2006), no. 1, 179-205 crossref(new window)

78.
K. H. Lee, Strongly pseudoconvex domains in almost complex manifolds, J. Reine Angew. Math. (To appear.)

79.
S. Lee, Asymptotic behavior of the Kobayashi metric on certain infinite-type pseudoconvex domains in $C^2$, J. Math. Anal. Appl. 256 (2001), no. 1, 190-215 crossref(new window)

80.
D. Ma, Sharp estimates of the Kobayashi metric near strongly pseudoconvex points, The Madison Symposium on Complex Analysis (Madison, WI, 1991), 329-338, Contemp. Math., 137, Amer. Math. Soc., Providence, RI, 1992

81.
X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Math. Acad. Sci. Paris 339 (2004), no. 7, 493-498 crossref(new window)

82.
X. Ma and G. Marinescu, Holomorphic Morse Inequalities and Bergman Kernels, Progress in Mathematics, 254. Birkhauser Verlag, Basel, 2007

83.
J. McNeal, Boundary behavior of the Bergman kernel function in $C^2$, Duke Math. J. 58 (1989), no. 2, 499-512 crossref(new window)

84.
J. McNeal, Local geometry of decoupled pseudoconvex domains, Complex analysis (Wuppertal, 1991), 223-230, Aspects Math., E17, Vieweg, Braunschweig, 1991

85.
J. McNeal, Estimates on the Bergman kernels of convex domains, Adv. Math. 109 (1994), no. 1, 108-139 crossref(new window)

86.
J. McNeal, Subelliptic estimates and scaling in the $\partial$-Neumann problem, Explorations in complex and Riemannian geometry, 197-217, Contemp. Math., 332, Amer. Math. Soc., Providence, RI, 2003 crossref(new window)

87.
J. Moser, Holomorphic equivalence and normal forms of hypersurfaces, Differential geometry (Proc. Sympos. Pure Math., Vol. XXVII, Part 2, Stanford Univ., Stanford, Calif., 1973), pp. 109-112. Amer. Math. Soc., Providence, R. I., 1975

88.
J. Moser, The holomorphic equivalence of real hypersurfaces, Proceedings of the International Congress of Mathematicians (Helsinki, 1978), pp. 659-668, Acad. Sci. Fennica, Helsinki, 1980

89.
J. Moser and S. Webster, Normal forms for real surfaces in $C^2$ near complex tangents and hyperbolic surface transformations, Acta Math. 150 (1983), no. 3-4, 255-296 crossref(new window)

90.
A. Nagel, J. P. Rosay, E. M. Stein, and S. Wainger, Estimates for the Bergman and Szego kernels in $C^2$, Ann. of Math. (2) 129 (1989), no. 1, 113-149 crossref(new window)

91.
R. Narasimhan, Several Complex Variables, University of Chicago Press, Chicago, IL, 1971

92.
L. Nirenberg, Lectures on linear partial differential equations, Amer. Math. Soc., Providence, RI, 1973

93.
S. Pinchuk, The scaling method and holomorphic mappings, Several complex variables and complex geometry, Part 1 (Santa Cruz, CA, 1989), 151-161, Proc. Sympos. Pure Math., 52, Part 1, Amer. Math. Soc., Providence, RI, 1991

94.
J. P. Rosay, Sur une caracterisation de la boule parmi les domaines de $C^n$ par son groupe d'automorphismes, Ann. Inst. Fourier (Grenoble) 29 (1979), no. 4, ix, 91-97 crossref(new window)

95.
R. Saerens and W. Zame, The isometry groups of manifolds and the automorphism groups of domains, Trans. Amer. Math. Soc. 301 (1987), no. 1, 413-429 crossref(new window)

96.
R. Schoen, On the conformal and CR automorphism groups, Geom. Funct. Anal. 5 (1995), no. 2, 464-481 crossref(new window)

97.
S. Sternberg, Local contractions and a theorem of Poincare, Amer. J. Math. 79 (1957), 809-824 crossref(new window)

98.
N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. (N.S.) 2 (1976), no. 1, 131-190

99.
S.Webster, On the Moser normal form at a non-umbilic point, Math. Ann. 233 (1978), no. 2, 97-102 crossref(new window)