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Foliations by complex manifolds involving the complex Hessian

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

Summary
In 1979 the second named author proved, in a joint paper with J. Ławrynowicz, the existence of a foliation of a bounded domain in $ℂ^n$ by complex submanifolds of codimension k+p-1, connected in some sense with a real (1,1) C³-form of rank k and the pth power of the complex Hessian of a C³-function u with im u plurisubharmonic and the property that for every leaf of this foliation the restricted functions im u, re u and $(∂/∂z_j) im u$, $(∂/∂z_j) re u$ are pluriharmonic and holomorphic, respectively.
Now the theorem is extended in two directions: to holomorphically decomposable (k,k)-forms, k < n, of class C³, and to exterior products of the complex Hessians of p plurisubharmonic C³-functions. This vast generalization gives rise to considerable extensions of the existence theorem of J. Ławrynowicz and M. Okada on a natural Markov process associated with the foliation as well as to the study of some of its properties. The main result is that the diffusion $X_t^θ$ uniquely determined by a foliation has the property that the sample paths of $X_t^θ$ remain to diffuse on leaves. Next, the convex case is examined and some examples depending on special and arbitrary holomorphic functions are presented. Since the foliations and canonical diffusions can be constructed in those cases effectively, we arrive at some properties of holomorphic functions on hypersurfaces, eliminating the inconvenient notions of foliations and canonical diffusions.
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.
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  • [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.
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Języki publikacji
EN
Uwagi
1991 Mathematics Subject Classification: Primary 32F05; Secondary 31C10, 60J45.
Identyfikator YADDA
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0012-3862
Kolekcja
DML-PL
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