Méthodes de changement d’échelles en analyse complexe

Berteloot, François

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 15 (2006) no. 3, pp. 427-483.

Nous mettons en perspective différentes méthodes de changement d’échelles et illustrons leur pertinence en mettant sur pieds des preuves simples et élémentaires de plusieurs théorèmes biens connus en analyse ou géométrie complexe. Les situations abordées sont variées et la plupart des théorèmes démontrés sont des classiques initialement obtenus entre la fin du xixe  et la seconde moitié du xxe  siècle.

We discuss several rescaling methods in complex analysis and geometry and apply them to get elementary proofs of some classical results. The Bloch principle plays an important role in our approach and yields to a somewhat unified point of view.

[1] Arnold, V. Chapitres supplémentaires de la théorie des équations différentielles ordinaires, Editions MIR Moscou - Editions du Globe Paris, 1996 | MR | Zbl

[2] Bargmann, D. Simple proofs of some fundamental properties of the Julia set, Ergodic Theory Dynam. Systems, Volume 19 (1999) no. 3, pp. 553-558 | MR | Zbl

[3] Bedford, E.; Pinchuk, S. Domains in ${\mathbf{C}}^2$ with non-compact automorphism group, Math. USSR Sbornik, Volume 63 (1989), pp. 141-151 | MR | Zbl

[4] Bedford, E.; Pinchuk, S. Convex domains with non-compact automorphism group, J. Geometric Anal., Volume 1 (1991), pp. 165-191 | MR | Zbl

[5] Bell, S. The Bergman kernel function and proper holomorphic mappings, Trans. Amer. Math. Soc., Volume 270 (1982) no. 2, pp. 685-691 | MR | Zbl

[6] Bell, S.; Ligocka, E. A simplification and extension of Fefferman’s theorem on biholomorphic mappings, Invent. Math., Volume 57 (1980) no. 3, pp. 283-289 | MR | Zbl

[7] Bergweiler, W. Rescaling principles in function theory, Analysis and its applications, Allied Publ., New Delhi (2001), pp. 11-29 (Chennai, 2000) | MR | Zbl

[8] Berteloot, F. Attraction de disques analytiques et continuité Höldérienne d’applications holomorphes propres, Topics in Compl.Anal., Banach Center Publ. (1995), pp. 91-98 | MR | Zbl

[9] Berteloot, F. Characterization of models in ${\mathbf{C}}^2$ by their automorphism group, Int. J. Math., Volume 5 (1994) no. 5, pp. 619-634 | MR | Zbl

[10] Berteloot, F. Principe de Bloch et estimations de la métrique de Kobayashi des domaines de ${\mathbf{C}}^2$, J. Geom. Anal., Volume 13 (2003) no. 1, pp. 29-37 | MR | Zbl

[11] Berteloot, F.; Cœuré, G. Domaines de ${\mathbf{C}}^2$, pseudoconvexes et de type fini ayant un groupe non-compact d’automorphismes, Ann. Inst. Fourier Grenoble, Volume 41 (1991) no. 1, pp. 77-86 | EuDML | Numdam | Zbl

[12] Berteloot, F.; Dupont, C. Une caractérisation des exemples de Lattès par leur mesure de Green Comment. Math. Helv. (à paraître) | Zbl

[13] Berteloot, F.; Duval, J. Une démonstration directe de la densité des cycles répulsifs dans l’ensemble de Julia, Complex analysis and geometry, Volume 188, Progr. Math., Basel, 2000, pp. 221-222 (Paris, 1997) | Zbl

[14] Berteloot, F.; Duval, J. Sur l’hyperbolicité de certains complémentaires, L’Enseignement Mathématique, Volume 47 (2001), pp. 253-267 | MR | Zbl

[15] Berteloot, F.; Loeb, J. J. Une caractérisation géométrique des exemples de Lattès de ${\mathbf{P}}^k$, Bull. Soc. Math. Fr., Volume 129 (2001) no. 2, pp. 175-188 | EuDML | Numdam | MR | Zbl

[16] Berteloot, F.; Mayer, V. Rudiments de dynamique holomorphe (Cours Spécialisés), Volume 7, Société Mathématique de France, EDP Sciences, Les Ulis, 2001 | MR | Zbl

[17] Briend, J.-Y.; Duval, J. Exposants de Liapounoff et distribution des points périodiques d’un endomorphisme de ${\mathbf{P}}^k$, Acta Math., Volume 182 (1999) no. 2, pp. 143-157 | MR | Zbl

[18] Briend, J.-Y.; Duval, J. Deux caractérisations de la mesure d’équilibre d’un endomorphisme de ${\mathbf{P}}^k$, Publ. Math. Inst. Hautes Études Sci., Volume 93 (2001), pp. 145-159 | EuDML | Numdam | MR | Zbl

[19] Brody, R. Compact manifolds and hyperbolicity, Trans. Amer. Math. Soc., Volume 235 (1978), pp. 213-219 | MR | Zbl

[20] Byun, J.; Gaussier, H.; Kim, K.-T. Weak-type normal families of holomorphic mappings in Banach spaces and characterization of the Hilbert ball by its automorphism group, J. Geom. Anal., Volume 12 (2002) no. 4, pp. 581-599 | MR | Zbl

[21] Catlin, D. Estimates of Invariant metrics on pseudoconvex domains of dimension two, Math. Z., Volume 200 (1989), pp. 429-466 | EuDML | MR | Zbl

[22] Christ, M. $C^{\infty }$ irregularity of the $\overline{\partial }$-Neumann problem for worm domains, J. Amer. Math. Soc., Volume 9 (1996) no. 4, pp. 1171-1185 | MR | Zbl

[23] Coupet, B. Precise regularity up to the boundary of proper holomorphic mappings, Ann. Scuola Norm. Sup. Pisa Cl. Sci., Volume (4) 20 (1993) no. 3, pp. 461-482 | EuDML | Numdam | MR | Zbl

[24] Coupet, B.; Gaussier, H.; Sukhov, A. Regularity of CR maps between convex hypersurfaces of finite type, Proc. Amer. Math. Soc., Volume 127 (1999) no. 11, pp. 3191-3200 | MR | Zbl

[25] Coupet, B.; Pinchuk, S.; Sukhov, A. On boundary rigidity and regularity of holomorphic mappings, Int. J. Math., Volume 7 (1996) no. 5, pp. 617-643 | MR | Zbl

[26] Coupet, B.; Sukhov, A. On CR mappings between pseudoconvex hypersurfaces of finite type in ${\mathbf{C}}^2$, Duke Math. J. Vol., Volume 88 (1997) no. 2, pp. 281-304 | MR | Zbl

[27] Demailly, J.-P. Variétés hyperboliques et équations différentielles algébriques, Gaz. Math., Volume 73 (1997), pp. 3-23 | MR | Zbl

[28] Demailly, J.-P.; ElGoul, J. Hyperbolicity of generic surfaces of high degree in projective 3-space, Amer. J. Math., Volume 122 (2000) no. 3, pp. 515-546 | MR | Zbl

[29] Dethloff, G.; Schumacher, G.; Wong, P. M. On the hyperbolicity of the complements of curves in algebraic surfaces, Duke Math. J., Volume 78 (1995), pp. 193-212 | MR | Zbl

[30] Diederich, K.; Fornaess, J. E. Proper holomorphic maps onto pseudoconvex domains with real analytic boundary, Ann. Math., Volume 110 (1979), pp. 575-592 | MR | Zbl

[31] Diederich, K.; Pinchuk, S. Proper holomorphic maps in dimension 2 extend, Indiana. Math. J., Volume 44 (1995), pp. 1089-1126 | MR | Zbl

[32] Dinh, T. C.; Sibony, N. Dynamique des applications d’allure polynomiale, J. Math. Pures Appl. (9), Volume 82 (2003) no. 4, pp. 367-423 | MR | Zbl

[33] Dixon, P. G.; Esterle, J. Michael’s problem and the Poincaré-Fatou-Bieberbach phenomenon, Bull. Amer. Math. Soc. (N.S.), Volume 15 (1986) no. 2, pp. 127-187 | MR | Zbl

[34] Efimov, A. M. A generalization of the Wong-Rosay theorem for the unbounded case, Sb. Math., Volume 186 (1995) no. 7, pp. 967-976 | MR | Zbl

[35] Eremenko, A. A Picard type theorem for holomorphic curves, Period. Math. Hungar., Volume 38 (1999) no. 1-2, pp. 39-42 | MR | Zbl

[36] Fefferman, C. The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math., Volume 26 (1974), pp. 1-65 | EuDML | MR | Zbl

[37] Fornaess, J. E.; Sibony, N. Construction of P.S.H. functions on weakly pseudoconvex domains, Duke Math. J., Volume 58 (1989), pp. 633-656 | MR | Zbl

[38] Fornaess, J. E.; Sibony, N. Complex Dynamics in higher dimensions, Complex potential theory (Montréal, PQ, 1993) (NATO ASI series Math. and Phys. Sci.), Volume 439, Kluwer Acad. Publ. (1994), pp. 131-186 | MR | Zbl

[39] Fornaess, J. E.; Sibony, N. Complex Dynamics in higher dimensions II, Ann. of Math. Studies, Volume 137 (1995) (Princeton Univ. Press, Princeton, NJ) | MR | Zbl

[40] Forstneric, F. An elementary proof of Fefferman’s theorem, Exposition. Math., Volume 10 (1992) no. 2, pp. 135-149 | MR | Zbl

[41] Forstneric, F.; Rosay, J.-P. Localization of the Kobayashi metric and the boundary continuity of proper holomorphic mappings, Math. Ann., Volume 110 (1987), pp. 239-252 | EuDML | MR | Zbl

[42] Françoise, J.-P. Géométrie analytique et systèmes dynamiques, Presses Universitaires de France, Paris, 1995 (Cours de troisième cycle) | MR

[43] Frankel, S. Complex geometry of convex domains that cover varieties, Acta Math., Volume 163 (1989) no. 1-2, pp. 109-149 | MR | Zbl

[44] Gaussier, H. Characterization of models for convex domains (preprint) | MR

[45] Graham, I. Boundary behavior of the Carathéodory and Kobayashi metrics on strongly pseudoconvex domains in $C^n$ with smooth boundary, Trans. Amer. Math. Soc., Volume 207 (1975), pp. 219-240 | MR | Zbl

[46] Gromov, M. Foliated plateau problem, part II : harmonic maps of foliations, GAFA, Volume 1 (1991) no. 3, pp. 253-320 | EuDML | MR | Zbl

[47] Henkin, G. An analytic polyhedron is not biholomorphic to a strictly pseudoconvex domain, Dokl. Akad. Nauk SSSR, Volume 210 (1973), pp. 1026-1029 | MR | Zbl

[48] Isaev, A.; Krantz, S. Domains with non-compact automorphism group : a survey, Adv. Math., Volume 146 (1999) no. 1, pp. 1-38 | MR | Zbl

[49] Jonsson, M.; Varolin, D. Stable manifolds of holomorphic diffeomorphisms, Invent. Math., Volume 149 (2002) no. 2, pp. 409-430 | MR | Zbl

[50] Kim, K.-T.; Krantz, S. Some new results on domains in complex space with non-compact automorphism group, J. Math. Anal. Appl., Volume 281 (2003) no. 2, pp. 417-424 | MR | Zbl

[51] Kobayashi, G. Hyperbolic complex spaces, Grundlehren der Mathematischen Wissenschaften, 318, Springer-Verlag, Berlin, 1998 | MR | Zbl

[52] Landau, E. Uber di Blochste Konstante und zwei verwandte Weltkonstanten, Math. Z., Volume 30 (1929), pp. 608-634 | EuDML | JFM | MR

[53] Lang, S. Introduction to complex hyperbolic manifolds, Springer Verlag, 1987 | MR | Zbl

[54] Ligocka, E. Some remarks on extension of biholomorphic mappings, Analytic functions (Lecture Notes in Math.), Springer, Berlin, Proc. Seventh Conf., Kozubnik, 1979 (1980), pp. 350-363 | MR | Zbl

[55] Lohwater, A. J.; Pommerenke, Ch. On normal meromorphic functions, Ann. Acad. Sci. Fenn. Ser. A I (1973) no. 550 | MR | Zbl

[56] Pinchuk, S. On proper holomorphic mappings of strictly pseudoconvex domains, Sib. Math. J., Volume 15 (1974), pp. 909-917 | Zbl

[57] Pinchuk, S. The scaling method ans holomorphic mappings, Proc. Sympos. Pure Math., Volume 52 (1991) no. 1, pp. 151-161 | Zbl

[58] Pinchuk, S.; Khasanov, S. Asymptotically holomorphic functions and their applications, Math. USSR-Sb., Volume 62 (1989) no. 2, pp. 541-550 | MR | Zbl

[59] Range, R. Holomorphic functions and integral representations in several complex variables, Graduate Texts in Mathematics, 108, Springer-Verlag, New York, 1986 | MR | Zbl

[60] Ros, A. The Gauss map of minimal surfaces, Differential Geometry-Valencia (2001), pp. 235-250 | MR | Zbl

[61] Rosay, J.-P. Sur une caractérisation de la boule parmi les domaines de $C^n$ par son groupe d’automorphismes, Ann. Inst. Fourier, Volume 29 (1979) no. 4, pp. 91-97 | EuDML | Numdam | MR | Zbl

[62] Rosay, J.-P.; Rudin, W. Holomorphic maps from $C^n$ to $C^n$, Trans. Amer. Math. Soc., Volume 310 (1988) no. 1, pp. 47-86 | MR | Zbl

[63] Ruelle, D. Elements of differentiable dynamics and bifurcation theory, Academic Press, Inc., Boston, MA, 1989 | MR | Zbl

[64] Schwick, W. Repelling periodic points in the Julia set, Bull. London Math. Soc., Volume 29 (1997) no. 3, pp. 314-316 | MR | Zbl

[65] Sibony, N. A class of hyperbolic manifolds, Ann. Math. Studies (1981) no. 100, pp. 357-372 | MR | Zbl

[66] Sibony, N. Dynamique des applications rationnelles de ${\mathbf{P}}^k$, Panor. Synthèses, Soc. Math. France, Paris, Dynamique et géométrie complexes (Lyon, 1997) (1999), pp. 97-185 | MR | Zbl

[67] Siu, Y. T.; Yeung, S. K. Hyperbolicity of the complement of a generic smooth curve of high degree in the complex projective plane, Invent. Math., Volume 124 (1996) no. 1-3, pp. 573-618 | MR | Zbl

[68] Stensones, B. A proof of the Michael conjecture (1998) (Preprint)

[69] Stensones, B. Fatou-Bieberbach domains with $C^{\infty }$-smooth boundary, Ann. of Math. (2), Volume 145 (1997) no. 2, pp. 365-377 | MR | Zbl

[70] Sternberg, S. Local contractions and a theorem of Poincaré, Amer. J. Math., Volume 79 (1957), pp. 809-824 | MR | Zbl

[71] Sukhov, A. On boundary regularity of holomorphic mappings, Mat. Sb., Volume 185 (1994), pp. 131-142 | MR | Zbl

[72] Webster, S. On the reflection principle in several complex variables, Proc. Amer. Math. Soc., Volume 71 (1978) no. 1, pp. 26-28 | MR | Zbl

[73] Wong, B. Characterization of the unit ball in $C^n$ by its automorphism group, Invent. Math., Volume 41 (1977) no. 3, pp. 253-257 | EuDML | MR | Zbl

[74] Zalcman, L. Normal families : new perspectives, Bull. Amer. Math. Soc., Volume 35 (1998), pp. 215-230 | MR | Zbl

[75] Zalcman, L. A heuristic principle in complex function theory, Amer. Math. Monthly, Volume 82 (1975) no. 8, pp. 813-817 | MR | Zbl