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Foliations by complex manifolds involving the complex Hessian

Seria
Rozprawy Matematyczne tom/nr w serii: 331 wydano: 1994
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Warianty tytułu
Abstrakty
EN

CONTENTS
Summary...................................................................................................................................................................3
1. Introduction and an outline of results....................................................................................................................5
2. Capacities on complex manifolds and the generalized complex Monge-Ampère equations..................................8
3. Foliations............................................................................................................................................................10
4. Proof of the existence theorem in the holomorphically decomposable case.......................................................12
5. Proof of the existence theorem in the exterior product case...............................................................................14
6. Natural Markov processes connected with the foliation $ℒ_{k+p-1}$.................................................................16
7. Properties of canonical diffusions.......................................................................................................................18
8. Laplace-Beltrami operator on Riemannian manifolds.........................................................................................21
9. Harmonic theory on compact complex manifolds................................................................................................23
10. Laplace-Beltrami operator as the generator of a canonical diffusion................................................................27
11. Laplace-Beltrami operator in the case of the sphere and the hyperboloid........................................................29
12. Complex Hessian involving convex functions....................................................................................................33
13. Some examples of applications.........................................................................................................................36
14. Hypersurfaces in ℂ³ depending on two holomorphic functions.........................................................................41
References.............................................................................................................................................................43
Miejsce publikacji
Warszawa
Copyright
Seria
Rozprawy Matematyczne tom/nr w serii: 331
Liczba stron
45
Liczba rozdzia³ów
Opis fizyczny
Dissertationes Mathematicae, Tom CCCXXXI
Daty
wydano
1994
otrzymano
1991-03-11
poprawiono
1993-03-30
poprawiono
1994-02-16
Twórcy
  • Institute of Mathematics, Polish Academy of Sciences, Narutowicza 56, 90-136 Łódź, Poland
  • Institute of Physics, University of Łódź, Pomorska 149/153, 90-236 Łódź, Poland
autor
  • Institute of Mathematics, Łódź Technical University, Al. Politechniki 11, 93-590 Łódź, Poland
autor
  • Department of Mathematics, College of General Education, Tôhoku University, Sendai 980, Japan
Bibliografia
  • [1] A. Andreotti and J. Ławrynowicz, On the generalized complex Monge-Ampère equation on complex manifolds and related questions, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 25 (1977), 943-948.
  • [2] A. Andreotti and J. Ławrynowicz, The generalized complex Monge-Ampère equation and a variational capacity problem, ibid., 949-955.
  • [3] E. Bedford and M. Kalka, Foliations and complex Monge-Ampère equations, Comm. Pure Appl. Math. 30 (1977), 543-571.
  • [4] E. Bedford and B. A. Taylor, Variational properties of the complex Monge-Ampère equation. II. Intrinsic norms, Amer. J. Math. 101 (1979), 1131-1166.
  • [5] R. Bott, Lectures on characteristic classes and foliations, in: Lectures on Algebraic and Differential Topology, R. Bott, S. Gitler and I. M. James (eds.), Lecture Notes in Math. 279, Springer, Berlin, 1972, 1-94.
  • [6] S. S. Chern, H. I. Levine and L. Nirenberg, Intrinsic norms on a complex manifold, in: Global Analysis, Papers in Honor of K. Kodaira, D. C. Spencer and S. Iynaga (eds.), Univ. of Tokyo Press and Princeton Univ. Press, Tokyo, 1969, 119-139; reprinted in: S.-S. Chern, Selected Papers, Springer, New York, 1978, 371-391.
  • [7] A. Crumeyrolle, Clifford Algebras and Spinor Structures, Math. Appl., Kluwer, Dordrecht, 1989.
  • [8] P. Dolbeault and J. Ławrynowicz, Holomorphic chains and extendability of holomorphic mappings, in: Deformations of Mathematical Structures. Complex Analysis with Physical Applications. Selected papers from the Seminar on Deformations, Łódź-Lublin 1985/87, J. Ławrynowicz (ed.), Kluwer, Dordrecht, 1989, 191-204.
  • [9] K. Fujii, Classical solutions of higher-dimensional nonlinear sigma models on spheres, Lett. Math. Phys. 10 (1985), 49-54.
  • [10] M. Fukushima, Dirichlet Forms and Markov Processes, Kodansha and North-Holland, 1980.
  • [11] M. Fukushima and M. Okada, On conformal martingale diffusions and pluripolar sets, J. Funct. Anal. 55 (1984), 377-388.
  • [12] M. Fukushima and M. Okada, On Dirichlet forms for plurisubharmonic functions, Acta Math. 159 (1987), 171-213.
  • [13] B. Gaveau et J. Ławrynowicz, Espaces de Dirichlet invariants biholomorphes et capacités associées, Bull. Acad. Polon. Sci. Sér. Sci. Math. 30 (1982), 63-69.
  • [14] B. Gaveau et J. Ławrynowicz, Intégrale de Dirichlet sur une variété complexe I, in: Séminaire Pierre Lelong-Henri Skoda (Analyse), Années 1980/81, Lecture Notes in Math. 919, Springer, Berlin, 1982, 131-166.
  • [15] B. Gaveau et P. Malliavin, Courbure des surfaces de niveau des fonctions holomorphes bornées dans la boule, C. R. Acad. Sci. Paris 293 (1981), 135-138.
  • [16] B. Gaveau, M. Okada and T. Okada, Explicit heat kernels on graphs and spectral analysis, manuscript.
  • [17] P. Griffiths and J. Harris, Principles of Algebraic Geometry, Wiley, New York, 1978.
  • [18] K. Itô and H. P. Mc Kean, Diffusion Processes and Their Sample Paths, Grundlehren Math. Wiss. 125, Springer, Berlin, 1965.
  • [19] J. Kalina, Biholomorphic invariance of the capacity and the capacity of an annulus, Ann. Polon. Math. 41 (1983), 175-184.
  • [20] J. Kalina and J. Ławrynowicz, Foliations and the generalized complex Monge-Ampère equations, in: Complex Analysis, Banach Center Publ. 11, PWN, Warszawa, 1983, 111-119.
  • [21] J. Kalina, J. Ławrynowicz, E. Ligocka and M. Skwarczyński, On some biholomorphic invariants in the analysis on manifolds, in: Analysis Functions, Kozubnik 1979, Proceedings, J. Ławrynowicz (ed.), Lecture Notes in Math. 798, Springer, Berlin, 1980, 224-249.
  • [22] S. Kanemaki and O. Suzuki, Hermitian pre-Hurwitz pairs and the Minkowski space, in: Deformations of Mathematical Structures. Complex Analysis with Physical Applications. Selected papers from the Seminar on Deformations, Łódź-Lublin 1985/87, J. Ławrynowicz (ed.), Kluwer, Dordrecht, 1989, 225-232.
  • [23] W. Królikowski, D. Lambert, J. Ławrynowicz and J. Rembieliński, The CR problem and Hurwitz pairs applied to field theory and solid state physics, Rep. Math. Phys. 29 (1990), 23-44.
  • [24] J. Ławrynowicz, Condenser capacities and an extension of Schwarz's lemma for hermitian manifolds, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. 23 (1975), 839-844.
  • [25] J. Ławrynowicz, On a class of capacities on complex manifolds endowed with an hermitian structure and their relation to elliptic and hyperbolic quasiconformal mappings, Dissertationes Math. 166 (1980), 49 pp.
  • [26] J. Ławrynowicz, Electromagnetic field and the theory of conformal and biholomorphic invariants, in: Complex Analysis and Its Applications III, International Atomic Energy Agency, 1976, 1-23.
  • [27] J. Ławrynowicz, On biholomorphic continuability of regular quasiconformal mappings, in: Analytic Functions, Kozubnik 1979, Proceedings, J. Ławrynowicz (ed.), Lecture Notes in Math. 798, Springer, Berlin, 1980, 326-349.
  • [28] J. Ławrynowicz and M. Okada, Canonical diffusion and foliation involving the complex hessian, Bull. Polish Acad. Sci. Math. 34 (1986), 661-667.
  • [29] H. B. Lawson, Foliations, Bull. Amer. Math. Soc. 80 (1974), 369-418.
  • [30] L. Nirenberg, A complex Frobenius theorem, in: Seminars on Analytic Functions I, Lecture Notes, Institute for Advanced Study, Princeton, N.J., 1957, 1-189.
  • [31] M. Okada, Potentiels Kähleriens et espaces de Dirichlet, C. R. Acad. Sci. Paris 292 (1981), 159-161.
  • [32] M. Okada, Symbolic calculus applied to convex functions and associated diffusions, in: Deformations of Mathematical Structures. Complex Analysis with Physical Applications. Selected papers from the Seminar on Deformations, Łódź-Lublin 1985/87, J. Ławrynowicz (ed.), Kluwer, Dordrecht, 1989, 319-329.
  • [33] A. Trautman, Solutions of the Maxwell and Yang-Mills equations associated with Hopf fiberings, Internat. J. Theoret. Phys. 16 (1977), 561-565.
  • [34] K. Yosida, Functional analysis, 2nd ed., Grundlehren Math. Wiss. 123, Springer, Berlin, 1968.
Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 32F05; Secondary 31C10, 60J45.
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0012-3862
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DML-PL
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