Universitatea Babeş-Bolyai Cluj-Napoca
Facultatea de Matematică şi Informatică
Ciclul de studii: Masterat
Domeniul: Matematica
Programul de studii: Analiză reală şi complexă (în limba engleză)
Limba de predare: Engleză

SYLLABUS

I. General data
Code	Subject	
MT263	Complex Functions of Several Variables	

Semester	Hours: C+S+L	Category	Status	
1	2+1+1		compulsory	

II. Full status faculty members
Name and	Scientific	Didactic	Chair	Type of activity	
surname	title	title		C	S	L	
KOHR Gabriela	Ph.D.	Prof.	Function Theory	*			

Associated faculty members
Name and	Scientific	Institution	Type of	Type of activity	
surname	title		position	C	S	L	

III. Course objectives
The aim of this course is to introduce and thoroughgoing study certain fundamental results in the theory of functions of several complex variables. The introduction in certain special topics of geometric function theory of several complex variables. The students will be involved in the research activity.  	

IV. Course contents
1. Holomorphic functions of several complex variables. General properties
2. Holomorphic mappings. General properties. Locally biholomorphic mappings and biholomorphic 
mappings. Automorphisms of bounded domains in Cⁿ.
3. Pluriharmonic functions and plurisubharmonic functions. 
4. Holomorphic extension. Domains of holomorphy. Holomorphic convexity. General properties. 
5. Pseudoconvex domains. Hartogs pseudoconvexity. Levi pseudoconvexity. General properties.
6. Special subclasses of biholomorphic mappings on the unit ball in the n-dimensional complex space: 
starlike, convex mappings and applications which have parametric representation. 
7. Introduction in the theory of subordination chains in Cⁿ. The generalized Loewner differential equation on the unit ball. Applications	

V. Bibliography
1. Kohr, G., Basic Topics in Holomorphic Functions of Several Complex Variables, Cluj University 
Press, Cluj-Napoca, 2003.
2. Graham, I., Kohr, G., Geometric Function Theory in One and Higher Dimensions, Marcel 
Dekker Inc, New York, 2003.
3. Scheidemann, V., Introduction to Complex Analysis in Several Variables, Birkhäuser Verlag, Basel-
Boston-Berlin, 2005.
4. Gunning, R.C., Introduction to Holomorphic Functions of Several Variables, vol.I. Function Theory, 
Wadsworth & Brooks/Cole, Monterey, CA, 1990.
5. Hörmander, L., An Introduction to Complex Analysis in Several Variables, Second Edition. North-
Holland Publ. Co., Amsterdam-London, 1973.
6. Chabat, B., Introduction à l’Analyse Complexe, vol. II, Edition MIR, Moscou, 1990. 
7. Krantz, S.G., Function Theory of Several Complex Variables, Reprint of the 1992 Edition, AMS 
Chelsea Publishing, Providence, Rhode Island, 2001. 
8. Narasimhan, R., Several Complex Variables, The University of Chicago Press, Chicago, 1971.
9. Range, M., Holomorphic Functions and Integral Representations in Several Complex Variables 
Springer-Verlag, New York, 1986.
10. Rudin, W., Function Theory in the Unit Ball of Cⁿ, Springer-Verlag, New York, 1980.	

VI. Thematic of didactic activities per weeks
Lectures.
Lecture 1.
The n-dimensional  complex space. Holomorphic functions. Equivalent definitions of holomorphy. The generalized Cauchy-Riemann equations. Integral representation of holomorphic functions on the polyidsc. Power series. 
[1, Chapter 1], [3, Chaper 1], [4, Chapters A, B], [8, Chapter 1], [9, Chapter 1]. 

Lecture 2.
The maximum modulus theorem. Shilov boundary. Schwarz’s lemma. The identity theorem for holomorphic functions. The open mapping theorem. Other properties of holomorphic functions. 
 [1, Chapter 1], [3, Chapter 1], [4, Chapter A], [6, Chapter 1], [8, Chapter 1], [9, Chapter 1], [5-Suplimentary bibliography, Chapter 1].

Lecture 3.
Sequences and series of holomorphic functions. Uniform convergence on compact sets. Weierstass’, Montel, Vitali’s and Hurwitz’s theorems. Uniqueness sets for the holomorphic functions. Examples. Topological and metric structures of the space H(Ω). 
 [1, Chapter 1], [3, Chapter 1], [8, Chapter 1], [9, Chapter 1], [5-Suplimentary bibliography, Chapter 1].

Lecture 4.
Holomorphic mappings. General properties. The maximum principle for holomorphic mappings. Locally biholomorphic mappings and biholomorphic mappings. Equivalence between univalence and biholomorphy in Cⁿ.  
 [1, Chapter 2], [3, Chapter 3], [9, Chapter 1], [7, Chapter 11]. 

Lecture 5.
Poincare’s result concerning the non-biholomorphically equivalence of the Euclidean unit ball and the unit polydisc in Cⁿ, n≥2. Cartan’s uniqueness results. Fatou’s example.  
 [1, Chapter 2], [3, Chapter 3], [8, Chapter 5], [6, Chapter 1], [9, Chapter 2].

Lecture 6.
The automorphisms of the Euclidean unit ball. The automorphisms of the unit polydisc. 
 [1, Chapter 2], [8, Chapter 5], [7, Chapter 11], [3, Chapter 3], [10, Chapter 2]. 

Lecture 7.
Domains of holomorphy. General properties. Holomorphic convexity. Equivalence between these notions. 
 [1, Chapter 3], [3, Chapter 7], [4, Chapter G], [6, Chapter 3], [8, Chapter 7], [9, Chapter 2], [5-Suplimentary bibliography, Chapter 2].

Lecture 8.
Pseudoconvex domains. Hartogs’ pseudoconvexity. General properties.
 [1, Chapter 3], [4, Chapter M], [6, Chapter 3], [7, Chapter 3], [9, Chapter 2].

Lecture 9.
Levi’s pseudoconvexity. General properties.	
 [1, Chapter 3], [4, Chapter R], [6, Chapter 3], [9, Chapter 2], [7, Chapter 3], [5-Suplimentary bibliography, Chapter 3]. 

Lecture 10.
The n-dimensional version of the Carathéodory class. Compactness of the class M. Subordination. Applications of subordination principle. 
 [1, Chapter 4], [2, Chapter 6].  

Lecture 11.
Starlike mappings. The analytical characterization of starlikeness on the Euclidean unit ball and the unit polydisc. Coefficient bounds. Examples of starlike mappings on the Euclidean unit ball and the unit polydisc. 
 [1, Chapter 4], [2, Chapters 6,7], [2-Suplimentary biblipgraphy, Chapters 3,4], [4-Suplimentary biblipgraphy, Chapters 1, 3, 6]. 

Lecture 12.
Convex mappings. The analytical characterization of convexity on the Euclidean unit ball and the unit polydisc. Coefficient bounds. Examples of convex mappings on the Eucliudean unit ball and the unit polydisc. 
 [1, Chapter 4], [2, Cap. 6,7], [2-Suplimentary bibliography, Chapters 3,4], [4-Suplimentary bibliography, Chapters 2, 4, 6]. 

Lecture 13.
Introduction to the theory of subordination chains in Cⁿ. The generalized Loewner differential equation on the unit ball in Cⁿ. Applications.
 [1, Chapter 4], [2, Chapters 8, 11], [3-Suplimentary bibliography, Chapter 2]. 

Lecture 14.
Biholomorphic mappings that have parametric representation on the unit ball in Cⁿ. The embeddings of biholomorphic mappings with parametric representation into univalent subordination chains. Examples.
[1, Chapter 4], [2, Chapters 8, 11], [3-Suplimentary bibliography, Chapter 2]. 

Seminar.
Seminar 1.
General properties of holomorphic functions of several complex variables. Applications of Cauchy’s integral representations on the unit polydisc.
 [1, Chapter 1], [4, Chapter A], [6, Chapter 1], [9, Chapter 1]. 

Seminar 2.
Applications of the maximum modulus theorem and Schwarz’s lemma for holomorphic functions of several complex variables. Shilov boundaries of the Euclidean unit ball and the unit polydisc. 
 [1, Chapter 1], [4, Chapter A], [6, Chapter 1], [9, Chapter 1]. 

Seminar 3.
Harmonic functions and pluriharmonic functions. General properties. 
 [8, Chapter 3], [4, Chapter K], [7, Chapter 2], [4, Chapter 3].

Seminar 4.
Subharmonic functions and plurisubharmonic functions. General properties.
 [1, Chapter 3], [4, Chapters J, K,Q], [7, Chapter 2], [8, Chapter 3], [9, Chapter 2]. 

Seminar 5.
Hartogs’ and Forelli’s theorems. Applications
 [1, Chapter 1], [5, Chapter 1], [6, Chapter 1]. 

Seminar 6.
Automorphisms of bounded domains in Cⁿ.  
 [1, Chapter 2], [8, Chapter 5], [6, Chapter 1]. 

Seminar 7.
Holomorphic extension. Hartogs’ theorems. Singularities of holomorphic functions in Cⁿ, n≥2. Examples.
 [1, Cap. 3], [3, Cap. 2, 4], [4, Cap. D], [5, Cap. 2], [6, Cap. 3], [8, Cap. 4]. 

Seminar 8.
Hurwitz’s theorem for biholomorphic mappings. Convexity of bounded domains in Cⁿ with boundary of class C².
 [1, Appendix E], [8, Chapter 5], [7, Chapter 3]. 

Seminar 9.
Holomorphically convex Reinhardt domains. Examples.
 [3, Chapter 7], [7, Chapter 2].

Seminar 10.
The Levi solution for tube domains.
 [1, Chapter 3], [7, Chapter 3], [5, Chapter 2]. 

Seminar 11.
The n-dimensional version of Jack’s, Miller’s and Mocanu’s lemma. Applications to the theory of differential subordinations in Cⁿ. 
[2-Suplimentary bibliography, Chapter 2], [3-Suplimentary bibliography, Chapter 2].

Seminar 12.
Sufficient conditions of starlikeness for locally biholomorphic mappings.The Roper and Suffridge extension operator. 
 [2, Capters 6, 11], [2- Suplimentary bibliography, Chapter 3].

Seminar 13.
Sufficient conditions of convexity for locally biholomorphic mappings. Starlike mappings of order α on the Euclidean unit ball. The connection between convexity and starlikenes of order ½. 
[2, Chapter 6], [2-Suplimentary bibliography, Chapter 3].

Seminar 14.
Univalence criteria obtained by the method of differential subordinations. Characterizations of starlikeness and spirallikeness of type α by using subordination chains. 
 [2, Chapter 8], [3-Suplimentary bibliography, Chapter 3].	

VII. Didactic methods used
•Interactive presentations, lectures, proofs, conversations. Presentations of alternative explanations. 
•Applications of the treated notions. Homework assignments.
•Direct answers to students. 	

VIII. Assessment
Final grade: 
• Oral exam at the end of semester: 70%
• Evaluation of homework assignments during the semester and active attendance of students to the seminar activity: 30%  
There are valid the official rules of the university concerning the attendance of students to teaching 
activities.	

IX. Additional references
1. Kaup, L., Kaup, B., Holomorphic Functions of Several Variables, Walter de Gruyter & Co., Berlin-
New York, 1983.
2. Kohr, G., Liczberski, P., Univalent Mappings of Several Complex Variables, Cluj University Press, 
Cluj-Napoca, 1998. 
3. Curt, P., Capitole Speciale de Teoria Geometricã a Funcţiilor de Mai Multe Variabile Complexe, Editura Albastrã, Cluj-Napoca, 2001.
4. Gong, S., Convex and Starlike Mappings in Several Complex Variables, Kluwer Acad. Publ., 
Dordrecht, 1998.
5. Della, Sala G., Saracco, A., Simioniuc, A., Tomassini, G., Lectures on Complex Analysis and 
Analytic Geometry, Lecture Notes. Scuola Normale Superiore di Pisa (New Series), 3, Edizioni della 
Normale, Pisa, 2006.
6. D’Angelo, J.P., Several Complex Variables and the Geometry of Real Hypersurfaces, CRC Press, Boca Raton, 1993. 
7. Grauert, H., Fritzsche, K., Several Complex Variables, Springer-Verlag, New York, Heidelberg, 
Berlin, 1976. 
8. W. Ebeling, Functions of Several Complex Variables and Their Singularities, American Math. Soc. 
Providence, Rhode Island, 2007. 
9. Mujica, J., Complex Analysis in Banach Spaces. Holomorphic Functions and Domains of 
Holomorphy in Finite and Infinite Dimensions, North-Holland, Amsterdam, 1986. 
10. K. Gürlebeck, K. Habetha, W. Spröβig, Holomorphic Functions in the Plane and n-Dimensional 
Space, Birkhäuser, Basel-Boston-Berlin, 2008.	

               Dean,	                                                                                  Chair head, 
Professor Leon Ţâmbulea, Ph.D.			                Professor Grigore Sãlãgean, Ph.D.

                                                                                                 						Date,		                                                                          Course responsible,
    March 3 2009					                Professor KOHR Gabriela, Ph.D.