Use of Invariant metric/distances to classify domains in $\mathbb{C}^n$

Question

I am a graduate student in mathematics, who works usually in operator theory. Lately I had to read about about the Lempert’s theorem(a theorem regarding when some pseudometric/distances coincide) and the domains for which the theorem holds. I wanted to know why actually does one study the same.

It is said on the Introduction of this thesis ‘In order to study domains in $n$-dimensional complex space $\mathbb{C}^n$, it is important to associate with these domains distance functions which are invariant under biholomorphic mappings. Such functions can then be used as a tool to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent.’

Again the Introduction in this thesis says that ‘Biholomorphically invariant metrics appear to be an important tool in Several Complex Variables which generalize the concept of Hermitian metrics. The Carath ́eodory and Kobayashi pseudometrics are instances of such metrics; both generalize the Poincar ́e metric and play an important role in the classification of domains’.

Can anyone tell ‘how’ actually are such invariant distances used as a tool to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent? Because everywhere they just mention the above and don’t actually explain how. Can you explain it in simple functional analysis terms? As I am not much familiar with differential/ hyperbolic geometry.

Answer 1

The best way to answer you is perhaps to directly quote part of the introduction of the beautiful book "Invariant distances and metrics in complex analysis" by M. Jarnicki and P. Pflug.

Here is the quote:

"One of the most beautiful results in the classical complex analysis is the Riemann mapping theorem which says that, except the whole complex plane, every simply connected plane domain is biholomorphically equivalent to the unit disc. Thus, the topological property "simply connected" is already sufficient to describe, up to biholomorphisms, a large class of plane domains. On the other hand, the Euclidean ball and the bidisc in $\mathbb C^2$ are topologically equivalent simply connected domains but they are not biholomorphic. This observation, which was made by H. Poincaré as early as at the end of the last century, shows that even inside the class of bounded simply connected domains there is no single model (up to biholomorphisms) as it is in the plane case. Therefore, it seems to be important to associate with domains in $\mathbb C^n$ tractable objects that are invariant under biholomorphic mappings. Provided that these objects are sufficiently concrete, one can hope to be able to decide, at least in principle, whether two given domains are biholomorphically distinct. An object of this kind was introduced, for example, by C. Caratheodory in the thirties. His main idea was to use the set of bounded holomorphic functions as an invariant. More precisely, he defined pseudodistances on domains via a "generalized" Schwarz Lemma. A specific property of these pseudodistances is that holomorphic mappings act as contractions. Thus, in particular, biholomorphic mappings operate as isometries. For such objects the name "invariant pseudodistances" has become very popular. This is where the title of our book comes from, although in the text we prefer to talk about holomorphically contractible pseudodistances. Apart from the class of bounded holomorphic functions, other classes of functions are used to obtain, via extremal problems, new objects contractible with respect to certain families of holomorphic mappings. For example, the class of square integrate holomorphic functions was used by S. Bergman. Moreover, all these objects admit infinitesimal versions associating to any "tangent vector" a specific length contractible under holomorphic mappings. Besides using families of functions to associate (via an extremal problem) tractable objects with domains in $\mathbb C^n$, one can consider sets of analytic discs as new biholomorphic invariants. This idea is due to S. Kobayashi."

Summing up, I would rather say that these invariant distances can be used as a tool to determine whether domains in $\mathbb C^n$ are biholomorphically inequivalent!

For instance, to biholomorphically distinguish the unit ball $\mathbb B^n$ in $\mathbb C^n$ and the polydisc $\Delta^n$ one can consider the Bergman metrics of these two domains, compute their holomorphic sectional curvatures and observe that it is constant for $\mathbb B^n$, while it is not constant for $\Delta^n$.