Foliations by complex manifolds involving the complex Hessian

Julian Ławrynowicz ; Jerzy Kalina ; Masami Okada

GDML_Books, (1994), p.

Résumé

SummaryIn 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.

CONTENTSSummary...................................................................................................................................................................31. Introduction and an outline of results....................................................................................................................52. Capacities on complex manifolds and the generalized complex Monge-Ampère equations..................................83. Foliations............................................................................................................................................................104. Proof of the existence theorem in the holomorphically decomposable case.......................................................125. Proof of the existence theorem in the exterior product case...............................................................................146. Natural Markov processes connected with the foliation $ℒ_{k+p-1}$.................................................................167. Properties of canonical diffusions.......................................................................................................................188. Laplace-Beltrami operator on Riemannian manifolds.........................................................................................219. Harmonic theory on compact complex manifolds................................................................................................2310. Laplace-Beltrami operator as the generator of a canonical diffusion................................................................2711. Laplace-Beltrami operator in the case of the sphere and the hyperboloid........................................................2912. Complex Hessian involving convex functions....................................................................................................3313. Some examples of applications.........................................................................................................................3614. Hypersurfaces in ℂ³ depending on two holomorphic functions.........................................................................41References.............................................................................................................................................................43

1991 Mathematics Subject Classification: Primary 32F05; Secondary 31C10, 60J45.

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Zbl 0808.32020

EUDML-ID : urn:eudml:doc:268588