Abate M., et al. Real methods in complex and CR geometry (LNM 1848, Springer,2004)(ISBN 3540223584)(223s)_PD_.pdf

Lecture Notes in Mathematics

1848

Editors:

J.--M. Morel, Cachan

F. Takens, Groningen

B. Teissier, Paris

Subseries:

Fondazione C.I.M.E., Firenze

Adviser: Pietro Zecca

M.Abate J.E.Fornaess X.Huang

J.-P. Rosay A. Tumanov

Real Methods in

Complex and CR

Geometry

Lecturesgivenatthe

C.I.M.E. Summer School

held in Martina Franca, Italy,

June

J u l y

2002

Editors: D. Zaitsev

G. Zampieri

123

Editors and Authors

Marco Abate

Department of Mathematics

University of Pisa

via Buonarroti 2

56127

Alexander Tumanov

Department of Mathematics

University of Illinois

1409

W. Green Street

Urbana,

Pisa Italy

e-mail: abate@dm.unipi.it

John Erik Fornaess

Department of Mathematics

University of Michigan

East Hall, Ann Arbor

MI 48109

, U.S.A.

e-mail: tumanov@math.uiuc.edu

Dmitri Zaitsev

School of Mathematics

Trinity College

University of Dublin

Dublin

IL 61801

, U.S.A.

e-mail: fornaess@umich.edu

Xiaojun Huang

Department of Mathematics

Rutgers University

New Brunswick

N.J. 08903

, Ireland

e-mail: zaitsev@maths.tcd.ie

Giuseppe Zampieri

Department of Mathematics

University of Padova

viaBelzoni7

35131

, U.S.A.

e-mail: huangx@math.rutgers.edu

Jean-Pierre Rosay

Department of Mathematics

University of Wisconsin

Madison,

Padova, Italy

e-mail: zampieri@math.unipd.it

, USA

e-mail: jrosay@math.wisc.edu

WI 53706-1388

LibraryofCongressControlNumber: 2004094684

Mathematics Subject Classification (2000):

32V05, 32V40, 32A40, 32H50 32VB25, 32V35

ISSN

0075-8434

ISBN

3-540-22358-4

Springer Berlin Heidelberg New York

DOI:

10.1007

98482

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Preface

The C.I.M.E. Session “Real Methods in Complex and CR Geometry” was held

in Martina Franca (Taranto), Italy, from June 30 to July 6, 2002. Lecture series

were given by:

M. Abate: Angular derivatives in several complex variables

J. E. Fornaess: Real methods in complex dynamics

X. Huang: On the Chern-Moser theory and rigidity problem for holomor-

phic maps

J. P. Rosay: Theory of analytic functionals and boundary values in the

sense of hyperfunctions

A. Tumanov: Extremal analytic discs and the geometry of CR manifolds

These proceedings contain the expanded versions of these ﬁve courses. In

their lectures the authors present at a level accessible to graduate students the

current state of the art in classical ﬁelds of the geometry of complex manifolds

(Complex Geometry) and their real submanifolds (CR Geometry). One of the

central questions relating both Complex and CR Geometry is the behavior

of holomorphic functions in complex domains and holomorphic mappings be-

tween diﬀerent complex domains at their boundaries. The existence problem

for boundary limits of holomorphic functions (called boundary values) is ad-

dressed in the Julia-Wolﬀ-Caratheodory theorem and the Lindelof principle

presented in the lectures of M. Abate. A very general theory of boundary val-

ues of (not necessarily holomorphic) functions is presented in the lectures of

J.-P. Rosay. The boundary values of a holomorphic function always satisfy the

tangential Cauchy-Riemann (CR) equations obtained by restricting the clas-

sical CR equations from the ambient complex manifold to a real submanifold.

Conversely, given a function on the boundary satisfying the tangential CR

equations (a CR function), it can often be extended to a holomorphic func-

tion in a suitable domain. Extension problems for CR mappings are addressed

in the lectures of A. Tumanov via the powerful method of the extremal and

stationary discs. Another powerful method coming from the formal theory and

Preface

inspired by the work of Chern and Moser is presented in the lectures of X.

Huang addressing the existence questions for CR maps. Finally, the dynamics

of holomorphic maps in several complex variables is the topic of the lectures

of J. E. Fornaess linking Complex Geometry and its methods with the theory

of Dynamical Systems.

We hope that these lecture notes will be useful not only to experienced

readers but also to the beginners aiming to learn basic ideas and methods in

these ﬁelds.

We are thankful to the authors for their beautiful lectures, all participants

from Italy and abroad for their attendance and contribution and last but not

least CIME for providing a charming and stimulating atmosphere during the

school.

Dmitri Zaitsev and Giuseppe Zampieri

CIME’s activity is supported by:

Ministero degli Aﬀari Esteri - Direzione Generale per la Promozione e la

Cooperazione - Ucio V;

Consiglio Nazionale delle Ricerche;

E.U. under the Training and Mobility of Researchers Programme.

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