# COMPLEX SCALING AND GEOMETRIC ANALYSIS OF SEVERAL VARIABLES

• Published : 2008.08.31

#### 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.

#### 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 https://doi.org/10.1007/BF02564462
4. E. Bedford and S. Pinchuk, Domains in $C^n+1$ with noncompact automorphism group, J. Geom. Anal. 1 (1991), no. 3, 165-191 https://doi.org/10.1007/BF02921302
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 https://doi.org/10.2307/1971379
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 https://doi.org/10.1007/BF01418930
8. F. Berteloot, Characterization of models in $C^2$ by their automorphism groups, Internat. J. Math. 5 (1994), no. 5, 619-634 https://doi.org/10.1142/S0129167X94000322
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. https://doi.org/10.4310/jdg/1214433979
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 https://doi.org/10.1307/mmj/1029005306
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 https://doi.org/10.1007/BF01390277
13. J. Byun, On the automorphism group of the Kohn-Nirenberg domain, J. Math. Anal. Appl. 266 (2002), no. 2, 342-356 https://doi.org/10.1006/jmaa.2001.7736
14. J. Byun, On the boundary accumulation points for the holomorphic automorphism groups, Michigan Math. J. 51 (2003), no. 2, 379-386 https://doi.org/10.1307/mmj/1060013203
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 https://doi.org/10.1016/j.crma.2005.09.018
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 https://doi.org/10.1007/BF02930654
17. D. Catlin, Estimates of invariant metrics on pseudoconvex domains of dimension two, Math. Z. 200 (1989), no. 3, 429-466 https://doi.org/10.1007/BF01215657
18. S. S. Chern and J. Moser, Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219-271 https://doi.org/10.1007/BF02392146
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 https://doi.org/10.2307/1990950
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 https://doi.org/10.1090/conm/332/05928
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 https://doi.org/10.2307/1971120
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 https://doi.org/10.1007/BF01406845
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 https://doi.org/10.1016/0001-8708(88)90002-3
30. S. Frankel, Complex geometry of convex domains that cover varieties, Acta Math. 163 (1989), no. 1-2, 109-149 https://doi.org/10.1007/BF02392734
31. S. Fu, Asymptotic expansions of invariant metrics of strictly pseudoconvex domains, Canad. Math. Bull. 38 (1995), no. 2, 196-206 https://doi.org/10.4153/CMB-1995-028-9
32. H. Gaussier and A. Sukhov, On the geometry of model almost complex manifolds with boundary, Math. Z. 254 (2006), no. 3, 567-589 https://doi.org/10.1007/s00209-006-0959-1
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 https://doi.org/10.24033/bsmf.2486
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 https://doi.org/10.2307/1997175
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 https://doi.org/10.1016/0001-8708(82)90028-7
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 https://doi.org/10.1007/BF01457445
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 https://doi.org/10.1007/BF01388806
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 https://doi.org/10.1007/BF02391775
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 https://doi.org/10.1080/03605309608821246
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 https://doi.org/10.1006/aima.1998.1821
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 https://doi.org/10.4153/CJM-2002-048-2
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 https://doi.org/10.1007/BF01444637
50. K.-T. Kim, On the automorphism groups of convex domains in Cn, Adv. Geom. 4 (2004), no. 1, 33-40 https://doi.org/10.1515/advg.2004.005
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 https://doi.org/10.2140/pjm.1992.155.99
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 https://doi.org/10.1090/S0002-9947-02-02895-7
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 https://doi.org/10.1515/form.2002.033
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 https://doi.org/10.4134/BKMS.2003.40.3.503
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 https://doi.org/10.1016/j.jmaa.2004.09.024
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 https://doi.org/10.1007/BF02921967
60. K.-T. Kim and G. Schmalz, Dynamics of local automorphisms of embedded CRmanifolds, Mat. Zametki 76 (2004), no. 3, 477-480 https://doi.org/10.4213/mzm575
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 https://doi.org/10.1023/B:MATN.0000043473.56503.f3
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 https://doi.org/10.2140/pjm.1996.176.141
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 https://doi.org/10.1512/iumj.1978.27.27020
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 https://doi.org/10.4310/jdg/1214430641
67. J. J. Kohn and L. Nirenberg, A pseudo-convex domain not admitting a holomorphic support function, Math. Ann. 201 (1973), 265-268 https://doi.org/10.1007/BF01428194
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 https://doi.org/10.1016/j.jmaa.2004.08.010
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 https://doi.org/10.2307/2007063
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 https://doi.org/10.1007/BF03322857
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 https://doi.org/10.1090/S0002-9947-05-03973-5
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 https://doi.org/10.1307/mmj/1144437443
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 https://doi.org/10.1006/jmaa.2000.7307
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 https://doi.org/10.1016/j.crma.2004.07.016
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 https://doi.org/10.1215/S0012-7094-89-05822-5
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 https://doi.org/10.1006/aima.1994.1082
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 https://doi.org/10.1090/conm/332/05937
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 https://doi.org/10.1007/BF02392973
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 https://doi.org/10.2307/1971487
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 https://doi.org/10.5802/aif.768
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 https://doi.org/10.2307/2000347
96. R. Schoen, On the conformal and CR automorphism groups, Geom. Funct. Anal. 5 (1995), no. 2, 464-481 https://doi.org/10.1007/BF01895676
97. S. Sternberg, Local contractions and a theorem of Poincare, Amer. J. Math. 79 (1957), 809-824 https://doi.org/10.2307/2372437
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 https://doi.org/10.4099/math1924.2.131
99. S.Webster, On the Moser normal form at a non-umbilic point, Math. Ann. 233 (1978), no. 2, 97-102 https://doi.org/10.1007/BF01421918
100. S.Webster, On the proof of Kuranishi's embedding theorem, Ann. Inst. H. Poincare Anal. Non Lineaire 6 (1989), no. 3, 183-207 https://doi.org/10.1016/S0294-1449(16)30322-5
101. J. Winkelmann, Realizing connected Lie groups as automorphism groups of complex manifolds, Comment. Math. Helv. 79 (2004), no. 2, 285-299 https://doi.org/10.1007/s00014-003-0794-5
102. B. Wong, Characterization of the unit ball in $C^n$ by its automorphism group, Invent. Math. 41 (1977), no. 3, 253-257 https://doi.org/10.1007/BF01403050
103. H.Wu, Old and new invariant metrics on complex manifolds, Several complex variables (Stockholm, 1987/1988), 640-682, Math. Notes, 38, Princeton Univ. Press, Princeton, NJ, 1993

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