Foliazioni di Monge-Ampère e classificazione olomorfa

Patrizio, Giorgio

Bollettino della Unione matematica italiana, Série 8, 8B (2005) no. 2, pp. 299-321.

Riassunto (VO)

Si illustrano alcuni sviluppi della teoria delle foliazioni di Monge-Ampère e delle sue applicazioni alla classificazione delle varietà complesse non compatte.

Abstract

We shall describe some developements of the theory of Monge-Ampère foliations and of its applications to the classification of non compact complex manifolds.

Bibliographie

[1] M. Abate - G. Patrizio , Finsler Metrics - A Global Approach. (with applications to geometric function theory), Springer Lecture Notes n. 1591 Springer Verlag, Berlin (1994). | MR | Zbl

[2] M. Abate - G. Patrizio , Finsler Metrics - Kèlher Finsler manifolds of constant holomorphic curvature, Int. J. of Math., 8 (1997), 169-186. | MR | Zbl

[3] R. Aguilar - D. Burns , On the algebraicization of certain Stein anifolds, Preprint 2001, arXiv:math.CV/0010287.

[4] L. V. Ahlfors , Conformal Invariants, Mc Graw Hill, New York (1973). | MR | Zbl

[5] A. F. Beardon , A Primer on Riemann Surfaces, Cambridge University Press, Cambridge (1984). | MR | Zbl

[6] E. Bedford - D. Burns , Holomorphic Mapping of Annuli in $\mathbb{C}^n$ and the Associated Extremal Function, Ann. Sc. Nor. Sup. Pisa, VI (1979), 381-414. | fulltext mini-dml | MR | Zbl

[7] E. Bedford - M. Kalka , Foliations and Complex Monge-Ampère Equations, Comm. Pure App. Math., 30 (1977), 543-571. | MR | Zbl

[8] E. Bedford - B. A. Taylor , A new capacity for plurisuharmonic functions, Acta Math., 149 (1982), 1-41. | MR | Zbl

[9] F. Berteloot - G. Patrizio , A Cartan theorem for proper holomorphic mappings of complete circular domains, Advances in Math., 153 (2000), 342-352. | MR | Zbl

[10] J. Bland - T. Duchamp , Moduli for Pointed Convex Domains, Invent. Math., 104 (1991), 61-112. | MR | Zbl

[11] D. Burns , Curvature of the Monge-Ampère foliation and parabolic manifolds, Ann. of Math., 115 (1982), 349-373. | MR | Zbl

[12] D. Burns , On the Uniqueness and Characterization of Grauert Tubes, In Complex Analysis and Geometry, Dekker Lecture Notes in Pure and Appl. Math., 173 (1995), 119-133. | MR | Zbl

[13] D. Burns and R. Hind , Symplectic geometry and the uniqueness of Grauert tubes, Geom. and Funct. Anal., 11 (2001), 1-10. | MR | Zbl

[14] D. Burns - S. Halverscheid - R. Hind , The geometry of Grauert tubes and complexification of symmetric spaces, Duke Math. J., 118 (2003), 465-491. | fulltext mini-dml | MR | Zbl

[15] H. Cartan , Les fonctions de deux variables complexes et le problème de représentation analytique, J. Math. Pures Appl., (9) 10 (1931), 1-114. | fulltext mini-dml

[16] J. P. Demailly , Mesures de Moge-Ampère et caractérizasatio géométrique des variétés algébriques affines, Bull. Soc. Math. France, Memoire, 19 (1985). | fulltext mini-dml | MR | Zbl

[17] T. Duchamp - M. Kalka , Singular Monge-Ampére foliations, Math. Ann., 325 (2003), 187-209. | MR | Zbl

[18] J. E. Fornaess - N. Sibony , Complex dynamics in higher dimensions, Complex Potential Theory, NATO ASI Series, Mathematical and Physical Sciences-Vol. 439 (1994), 131-186. | MR | Zbl

[19] V. Guillemin - M. Stenzel , Grauert Tubes and the Homogeneous Monge-Ampère Equation, J. of Diff. Geometry, 34 (1991), 561-570. | fulltext mini-dml | MR | Zbl

[20] H. Grauert , On Levi's problem and the imbedding of real-analytic manifolds, Ann. Math., 68 (1958), 460-472. | MR | Zbl

[21] R. E. Greene - G. Krantz , Deformation of Complex Structures, estimates of $\bar\partial$ Equation and Stabilty of the Bergmann Kernel, Adv. of Math., 43 (1982), 1-86. | MR | Zbl

[22] S. Halverscheid , Maximal domains of definiton of adapted complex structures of symmetric spaces of non-compact type, Schriftenreihe des Graduiertenkollegs Geometrie und Mathematische Physik, 39 (2001) Bochum: Univ. Bochum, Institut für Mathematik. | Zbl

[23] S. Halverscheid - A. Iannuzzi , Maximal complexifications of certain homogeneous Riemannian manifolds, Trans. Am. Math. Soc., 355 (2003), 4581-4594. | MR | Zbl

[24] A. T. Huckleberry - J. A. Wolf , Cycles Spaces of Flag Domains: A Complex Geometric Viewpoint, Preprint 2002, arXiv:math.RT/0210445. | fulltext mini-dml | MR | Zbl

[25] M. Kalka - G. Patrizio , Polynomial Solutions of the Complex Homogeneous Monge-Ampère Equation, Michigan Math. J., 52 (2004), 243-251. | fulltext mini-dml | MR | Zbl

[26] Su-Jen Kan , On rigidity of Grauert tubes over homogeneous Riemannian manifolds, To appear in J. reine angew. Math. | fulltext mini-dml | MR | Zbl

[27] M. Klimek , Pluripotential Theory, London Mathematical Society Monographs New Series, 6, Oxford (1991). | MR | Zbl

[28] P. Koebe , Über die Uniformisierung beliebiger analityscher Kurven, Göttinger Nachr. (1907), 191-210 e 633-669.

[29] L. Lempert , La métrique de Kobayashi et la représantation des domaines sur la boule, Bull. Soc. Math. France, 109 (1989), 427-474. | fulltext mini-dml | MR | Zbl

[30] L. Lempert , Complex Structures on the Tangent Bundle of Riemannian Manifolds, In Complex Analysis an and geometry, Plenum Press, (1993), 235-251. | MR | Zbl

[31] L. Lempert , Holomorphic invariants, normal forms, and the moduli space of convex domains, Annals of Math., 128 (1988), 43-78. | MR | Zbl

[32] L. Lempert , R. Szöke , Global Solutions of the Homogeneous Complex Monge-Ampère Equation and Complex Structures on the Tangent Bundle of Riemannian Manifolds, Math. Ann., 290 (1991), 689-712. | MR | Zbl

[33] Y. Matsushima , On tube domains, In Symmetric Spaces, short Courses presented at Washington Univ., Pure appl. Math. 8, Dekker, New York (1972), 255-270. | MR | Zbl

[34] G. Patrizio Parabolic Exhaustions for Strictly Convex Domains, Manuscripta Math., 47 (1984), 271-309. | MR | Zbl

[35] G. Patrizio , A Characterization of Complex Manifolds Biholomorphic to a Circular Domain, Math. Z., 189 (1985), 343-363. | MR | Zbl

[36] G. Patrizio - P. M. Wong , Stability of the Monge-Amère Foliation, Math. Ann., 263 (1983), 13-29. | MR | Zbl

[37] G. Patrizio - P. M. Wong , On Stein Manifolds with Compact Symmetric Center, Math. Ann., 289 (1991), 355-382. | MR | Zbl

[38] G. Patrizio - A. Tomassini , A Characterization of Affine Hyperquadrics, To appear in Annali di Matematica Pura e App. | Zbl

[39] H. Poincaré , Les fonctions analytiques de deux variables et la représantation conforme, Rend.Circ. Matem. Palermo, 23 (1907), 185-220.

[40] H. Poincaré , Sur l'uniformisation des fonctions analytiques, Acta Math., 31 (1908), 1-63. | MR

[41] W. Stoll , The Characterization of Strictly Parabolic Manifolds, Ann. Sc. Nor. Sup. Pisa, VII (1980), 87-154. | fulltext mini-dml | MR | Zbl

[42] R. Szöke , Complex Structures on the Tangent Bundle of Riemannian Manifolds, Math. Ann., 291 (1991), 409-428. | MR | Zbl

[43] R. Szöke , Adapted Complex Structures and Riemannian homogeneous spaces, Ann. Pol. Math., LXX (1998), 215-220. | MR | Zbl

[44] B. Totaro , Complexifications of nonnegatively curved manifolds, J. Eur. Math. Soc. (JEMS), 5 (2003), 69-94. | MR | Zbl

[45] E. Vesentini , Variations on a theme of Carathéodory, Ann. Sc. Nor. Sup. Pisa, VI (1979), 39-68. | fulltext mini-dml | MR | Zbl

[46] E. Vesentini , Complex geodesics, Comp. Math., 44 (1981), 375-394. | fulltext mini-dml | MR | Zbl

[47] H. Whitney - F. Bruhat , Quelques propriétés fondamentales des ensembles analytiques-réels, Comment. Math. Helv., 22 (1959), 132-160. | MR | Zbl

[48] P. M. Wong , Geometry of the homogeneous complex Monge-Ampère equation, Inv. Math., 67 (1982), 261-274. | MR | Zbl

[49] P. M. Wong , On Umbilical Hypersurfaces and Uniformization of Circular Domains, Proc. Symp. in Pure Math., 41 (1984), 225-252. | MR | Zbl