Why do we study biholomorphically invariant pseudodistances/metrics

Question

It is said that pseudodistances/metrics which are invariant under biholomorphic maps are used to determine whether domains in $\mathbb{C}^n$ are biholomorphically equivalent or not.

Suppose $\Omega_1$ and $\Omega_2$ are domains in $\mathbb{C}^n$, and $\rho_1$ and $\rho_2$ are pseudodistances on it such that $\rho_2(f(a),f(b))=\rho_2(a,b)$ for every biholomorphic mapping $f:\Omega_1\longrightarrow\Omega_2$. Then how exactly does one use the above fact to show that the two domains are biholomorphically equivalent or not? Do we take one map at a time and check if it is a isometry under these pseudodistances? But isnt that a tedious process and inefficient?

I also wanted to know as to why do we study domains on which certain pseudodistances are equal? Like the Lempert's Theorem states that the Caratheodory-Reiffen Pseudometric is equal to the Kobayashi-Royden pseudometric on Convex domains.

Answer 1

Invariant metrics and pseudometrics are used to study holomorphic maps between complex manifolds. The model is the use of the hyperbolic metric in dimension 1.

For example, a manifold is called hyperbolic if its Kobayashi pseudometric is a true metric. Then every holomorphic map from a line to a hyperbolic manifold is constant. This is a generalization of Picard's Little Theorem. Remarkably, for compact complex manifolds there is also the converse statement (Brody's Lemma).

There are also generalizations of Picard's Great Theorem, normality criteria and generalizations of removable singularity theorem. One major application is to the proof that the group of automorphisms of a compact hyperbolic space is finite.

All this has a lot of further applications, though much less than for the hyperbolic metric in one-dimensional case.

I recommend an excellent survey by Kobayashi himself, as well as his books:

Hyperbolic manifolds and holomorphic mappings, 1970 and Hyperbolic complex spaces, 1998, and the book by S. Lang, Introduction to complex hyperbolic spaces.

Answer 2

Just as in Riemannian (and Finsler) geometry, we can often distinguish domains by metric information, such as determining the behaviour of geodesics, or finding some analogue of curvature to compute, so we can often distinguish domains by such data arising from invariant metrics. For example, we can distinguish the ball from a product of lower dimensional balls, since we can find explicitly the Kobayashi metric of the ball to be the complex hyperbolic metric (which is Riemannian) and see that its curvature is not that of a product metric.

Such a procedure is not possible for complicated domains, for which we don't really know the metric, so we can't always do this. But that is a familiar phenomenon in mathematics.