# Foliations by complex manifolds involving the complex Hessian

Julian Ławrynowicz; Jerzy Kalina; Masami Okada

- Publisher: Instytut Matematyczny Polskiej Akademi Nauk(Warszawa), 1994

## Access Full Book

top## Abstract

top## How to cite

topJulian Ławrynowicz, Jerzy Kalina, and Masami Okada. Foliations by complex manifolds involving the complex Hessian. Warszawa: Instytut Matematyczny Polskiej Akademi Nauk, 1994. <http://eudml.org/doc/268588>.

@book{JulianŁawrynowicz1994,

abstract = {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.............................................................................................................................................................431991 Mathematics Subject Classification: Primary 32F05; Secondary 31C10, 60J45.},

author = {Julian Ławrynowicz, Jerzy Kalina, Masami Okada},

keywords = {foliation; Monge-Ampère operator; Dirichlet form; diffusion; Laplace- Beltrami operator},

language = {eng},

location = {Warszawa},

publisher = {Instytut Matematyczny Polskiej Akademi Nauk},

title = {Foliations by complex manifolds involving the complex Hessian},

url = {http://eudml.org/doc/268588},

year = {1994},

}

TY - BOOK

AU - Julian Ławrynowicz

AU - Jerzy Kalina

AU - Masami Okada

TI - Foliations by complex manifolds involving the complex Hessian

PY - 1994

CY - Warszawa

PB - Instytut Matematyczny Polskiej Akademi Nauk

AB - 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.............................................................................................................................................................431991 Mathematics Subject Classification: Primary 32F05; Secondary 31C10, 60J45.

LA - eng

KW - foliation; Monge-Ampère operator; Dirichlet form; diffusion; Laplace- Beltrami operator

UR - http://eudml.org/doc/268588

ER -

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.