Contents
Introduction
Chapter 1. Lempert functions and Kobayashi metrics
1.1. Lempert functions and their ”derivatives”
1.2. Balanced domains
1.3. Kobayashi–Buseman metric
1.4. Interpolation in the Arakelian theorem
1.5. Generalized Lempert function
1.6. Product property
Chapter 2. The symmetrized polydisc and the spectral ball
2.1. Preliminaries
2.2. Cyclic matrices
2.3.
G
n
is not a Lu Qi-Keng domain for n ≥ 3
2.4. Generalized balanced domains
2.5. Notions of complex convexity
2.6.
G
n
∈ L for n ≥ 3
2.7. Estimates for γ
G
2n+1
(0; e
2
)
2.8. Continuity of l
Ω
n
(A, ·)
2.9. Zeroes of κ
Ω
n
2.10. The Kobayashi metric vs. the Lempert function
Chapter 3. Estimates and boundary behavior of invariant metrics
on
C-convex domains
3.1. Estimates for the Carath´eodory and Kobayashi metrics
3.2. Types of boundary points
3.3. Estimates for the Bergman kernel and the Bergman metric 93
3.4. Maximal basis. A counterexample
3.5. Estimates in a maximal basis
3.6. Localizations
3.7. Localization of the Bergman kernel and the Bergman
metric
3.8. Boundary behavior of invariant metrics of planar domains 115
Bibliography
3