Invariant metrics: Carathéodory, Kobayashi, Bergman

Intuition Beginner

A holomorphic map cannot stretch distances. On the unit disc this is the content of the Schwarz lemma: a holomorphic self-map of the disc moves no pair of points farther apart than they already are, when you measure with the right hyperbolic ruler. The remarkable fact is that this ruler is forced on you. Once you insist that holomorphic maps never increase distance, and that the disc looks the same from every point, only one ruler survives.

In several variables there is no single domain as symmetric as the disc, so there is no single ruler. Instead each domain carries its own ruler, built so that every holomorphic map into or out of the domain shrinks distances. Two domains that holomorphic maps cannot tell apart wear identical rulers, so the ruler is a fingerprint of the domain's complex-analytic shape.

There are three natural ways to build such a ruler, named for Carathéodory, Kobayashi, and Bergman. They start from different recipes but share one defining property: holomorphic maps shrink them.

Visual Beginner

A bounded blob-shaped domain in the plane, with a small grid of unit-ruler circles drawn inside it. Near the centre the circles are large and round; as they approach the wrinkly boundary they shrink and squash, showing that the same true distance corresponds to ever-smaller Euclidean steps near the edge. An inset shows two such blobs related by a holomorphic map, an arrow between them, with a note that the map carries small ruler-circles to no-larger ruler-circles — the distance-shrinking rule. A third inset shows the round ball, where all three named rulers coincide into one perfectly symmetric pattern.

Worked example Beginner

Take the unit disc and measure the distance from the centre $0$ to a point $r$ on the positive axis, with $0<r<1$, using the hyperbolic ruler. The recipe assigns the number

$$\rho (0,r)=\frac{1}{2}\log \frac{1+r}{1-r}.$$

Try $r=0.5$. Then $(1+r)/(1-r)=1.5/0.5=3$, and the distance is $\frac{1}{2}\log 3\approx 0.549$. Now try $r=0.9$: the fraction is $1.9/0.1=19$, and the distance is $\frac{1}{2}\log 19\approx 1.472$. Push to $r=0.99$: the fraction is $199$, distance $\frac{1}{2}\log 199\approx 2.646$.

The Euclidean gap from $0.9$ to $0.99$ is only $0.09$, yet the hyperbolic distance grew by more than a full unit. As $r$ climbs toward $1$, the distance grows without bound: the boundary is infinitely far away.

What this tells us: the natural ruler on the disc puts the edge at infinite distance, so the disc is complete and a point can never reach the boundary in finitely many ruler-steps. The three named rulers all reduce to exactly this one on the disc.

Check your understanding Beginner

Formal definition Intermediate+

Let $\Delta =\{\zeta \in \mathbb{C}:|\zeta |<1\}$ carry the Poincaré distance

$$\rho ({\zeta }_1,{\zeta }_2)={\tanh }^{-1}\left|\frac{{\zeta }_1-{\zeta }_2}{1-{\bar{\zeta }}_1{\zeta }_2}\right|,$$

the distance function of the metric $\frac{|d\zeta |}{1-|\zeta {|}^2}$. Write $\mathrm{Hol}(X,Y)$ for the holomorphic maps $X\to Y$. Throughout, $\Omega \subseteq {\mathbb{C}}^n$ is a domain and $p,q\in \Omega$.

Definition (Carathéodory pseudodistance). The Carathéodory pseudodistance is

$$c_{\Omega }(p,q)=\sup \{\rho (f(p),f(q)):f\in \mathrm{Hol}(\Omega ,\Delta )\}.$$

It is the largest separation any bounded holomorphic function can produce, measured downstream in the disc. It is symmetric and satisfies the triangle inequality (inherited from $\rho$), but may fail $c_{\Omega }(p,q)=0\Rightarrow p=q$, hence pseudodistance.

Definition (Kobayashi pseudodistance). An analytic disc in $\Omega$ is a map $\phi \in \mathrm{Hol}(\Delta ,\Omega )$. For $p,q\in \Omega$ a chain from $p$ to $q$ is a finite sequence of analytic discs ${\phi }_1,\dots ,{\phi }_m$ and points $a_i,b_i\in \Delta$ with ${\phi }_1(a_1)=p$, ${\phi }_m(b_m)=q$, and ${\phi }_i(b_i)={\phi }_{i+1}(a_{i+1})$. The Kobayashi pseudodistance is

$$k_{\Omega }(p,q)=\inf \left\{\sum _{i=1}^m\rho (a_i,b_i):\text{chains from }p\text{ to }q\right\}.$$

It is the cheapest way to walk from $p$ to $q$ by hopping along discs, paying the Poincaré cost of each hop.

Definition (infinitesimal metrics). For $p\in \Omega$ and a tangent vector $v\in {\mathbb{C}}^n$, the infinitesimal Carathéodory and Kobayashi metrics are

$${\gamma }_{\Omega }(p;v)=\sup \{|d{f}_p(v)|:f\in \mathrm{Hol}(\Omega ,\Delta ),f(p)=0\},$$ $${\kappa }_{\Omega }(p;v)=\inf \{|\alpha {|}^{-1}:\phi \in \mathrm{Hol}(\Delta ,\Omega ),\phi (0)=p,\alpha {\phi }^{\prime }(0)=v,\alpha >0\},$$

the latter the reciprocal of the largest disc-radius achieving the direction $v$. The Bergman metric ${\beta }_{\Omega }$ is the metric of 06.10.08, $g_{j\bar{k}}={\partial }_{z_j}{\partial }_{{\bar{z}}_k}\log K_{\Omega }(z,z)$; it is the third invariant metric, defined whenever $\Omega$ is bounded so that $K_{\Omega }(z,z)>0$.

Counterexamples to common slips

• $c_{\Omega }$ and $k_{\Omega }$ are genuine pseudodistances on general domains: on $\Omega ={\mathbb{C}}^n$ they vanish identically, since Liouville forces every bounded holomorphic function to be constant (Carathéodory) and the whole plane $\mathbb{C}\to {\mathbb{C}}^n$ is an analytic "disc" of infinite radius (Kobayashi).
• The Bergman metric is not generally comparable to a constant multiple of ${\kappa }_{\Omega }$; the three metrics agree only on highly symmetric domains. On the bidisc the Bergman metric is a product metric, while $\kappa$ is the maximum of the factor metrics.
• The Carathéodory and Kobayashi distances are not the integrated lengths of $\gamma$ and $\kappa$ for $c$ in general (the Carathéodory distance can exceed the integrated infinitesimal metric), though Royden's theorem repairs this on the Kobayashi side.

Key theorem with proof Intermediate+

Theorem (distance-decreasing property; Schwarz-Pick in several variables). Let ${\Omega }_1\subseteq {\mathbb{C}}^n$ and ${\Omega }_2\subseteq {\mathbb{C}}^m$ be domains and $F\in \mathrm{Hol}({\Omega }_1,{\Omega }_2)$. Then for all $p,q\in {\Omega }_1$,

$$c_{{\Omega }_2}(F(p),F(q))\le c_{{\Omega }_1}(p,q),k_{{\Omega }_2}(F(p),F(q))\le k_{{\Omega }_1}(p,q).$$

In particular both pseudodistances are biholomorphic invariants: a biholomorphism is an isometry for $c$ and for $k$.

Proof. The two halves use the two extremal constructions in opposite directions; the engine in both is the one-variable Schwarz-Pick lemma 06.01.12, which states that every $g\in \mathrm{Hol}(\Delta ,\Delta )$ satisfies $\rho (g(a),g(b))\le \rho (a,b)$.

Carathéodory half. Let $f\in \mathrm{Hol}({\Omega }_2,\Delta )$ be any competitor for $c_{{\Omega }_2}(F(p),F(q))$. The composite $f\circ F\in \mathrm{Hol}({\Omega }_1,\Delta )$ is a competitor for $c_{{\Omega }_1}(p,q)$, so

$$\rho (f(F(p)),f(F(q)))=\rho ((f\circ F)(p),(f\circ F)(q))\le c_{{\Omega }_1}(p,q).$$

Taking the supremum over all $f\in \mathrm{Hol}({\Omega }_2,\Delta )$ on the left gives $c_{{\Omega }_2}(F(p),F(q))\le c_{{\Omega }_1}(p,q)$. No Schwarz-Pick step is needed here beyond the very definition of $\rho$; the invariance is purely formal, which is why the Carathéodory distance is the largest invariant pseudodistance bounded above by Poincaré pullbacks.

Kobayashi half. Let ${\phi }_1,\dots ,{\phi }_m$ with points $a_i,b_i$ be a chain from $p$ to $q$ in ${\Omega }_1$. Post-composing each disc with $F$ gives analytic discs $F\circ {\phi }_i\in \mathrm{Hol}(\Delta ,{\Omega }_2)$, and since $F\circ {\phi }_i(b_i)=F({\phi }_i(b_i))=F({\phi }_{i+1}(a_{i+1}))=F\circ {\phi }_{i+1}(a_{i+1})$, the family $(F\circ {\phi }_i)$ is a chain from $F(p)$ to $F(q)$ in ${\Omega }_2$ with the same parameter points $a_i,b_i$. Hence

$$k_{{\Omega }_2}(F(p),F(q))\le \sum _{i=1}^m\rho (a_i,b_i).$$

Taking the infimum over all chains in ${\Omega }_1$ on the right yields $k_{{\Omega }_2}(F(p),F(q))\le k_{{\Omega }_1}(p,q)$.

Invariance. If $F$ is a biholomorphism, apply the inequality to $F$ and to $F^{-1}$; the two opposite inequalities force equality. The same argument applies to the infinitesimal metrics, post-composing $d{f}_p$ (Carathéodory) and using $d{F}_p\circ {\phi }^{\prime }(0)$ (Kobayashi). $\square$

A second structural theorem orders the two distances.

Theorem (Carathéodory–Kobayashi comparison). For every domain $\Omega \subseteq {\mathbb{C}}^n$ and all $p,q\in \Omega$, $c_{\Omega }(p,q)\le k_{\Omega }(p,q)$, and likewise ${\gamma }_{\Omega }\le {\kappa }_{\Omega }$ pointwise.

Proof. Fix $p,q$ and a chain ${\phi }_1,\dots ,{\phi }_m$ with parameters $a_i,b_i$ realising a value close to $k_{\Omega }(p,q)$. Let $f\in \mathrm{Hol}(\Omega ,\Delta )$ be any Carathéodory competitor. Each $f\circ {\phi }_i\in \mathrm{Hol}(\Delta ,\Delta )$, so by one-variable Schwarz-Pick $\rho (f{\phi }_i(a_i),f{\phi }_i(b_i))\le \rho (a_i,b_i)$. The triangle inequality for $\rho$ along the chain, using $f{\phi }_i(b_i)=f{\phi }_{i+1}(a_{i+1})$, gives

$$\rho (f(p),f(q))\le \sum _{i=1}^m\rho (f{\phi }_i(a_i),f{\phi }_i(b_i))\le \sum _{i=1}^m\rho (a_i,b_i).$$

Taking the supremum over $f$ on the left and the infimum over chains on the right gives $c_{\Omega }(p,q)\le k_{\Omega }(p,q)$. The infinitesimal statement is the one-disc, one-point version of the same computation. $\square$

Bridge. The comparison $c_{\Omega }\le k_{\Omega }$ builds toward the rigidity theory that appears again in 06.10.13: a domain on which the two extremes coincide is forced to be unusually symmetric, and this is exactly the mechanism behind Lempert's theorem that $c=k$ on convex domains. The foundational reason both constructions are biholomorphic invariants is the one-variable Schwarz-Pick lemma 06.01.12, applied downstream for Carathéodory and upstream for Kobayashi; putting these together, the entire several-variable invariant-metric theory is the statement that the disc, with its Poincaré metric, is the universal yardstick, and every domain inherits an upper and a lower invariant reading from it. The central insight is that $c_{\Omega }$ is the largest invariant distance one can read off bounded holomorphic functions and $k_{\Omega }$ is the smallest invariant distance dominating analytic discs, so any invariant distance with the distance-decreasing property is squeezed between them; the Bergman metric of 06.10.08 is a third invariant living in this same squeeze on bounded domains, and the bridge is the universal role of the Poincaré disc.

Exercises Intermediate+

Lean formalization Intermediate+

Mathlib does not formalise the invariant-distance constructions of several complex variables: there is no named Poincaré distance on the disc as a metric-space instance, no space $\mathrm{Hol}(\Omega ,\Delta )$ as a topological object, and neither extremal construction. A proposed signature, in Lean 4 / Mathlib syntax, sketching the target statements:

-- Sketch only; no current Mathlib coverage. See lean_mathlib_gap.
import Mathlib.Analysis.Complex.Circle
import Mathlib.Topology.MetricSpace.Basic

namespace Codex.SeveralComplexVariables.InvariantMetrics

variable {n m : ℕ}
variable (Ω : Set (Fin n → ℂ)) (Ω' : Set (Fin m → ℂ))

-- Poincaré distance on the unit disc (placeholder signature).
noncomputable def poincare (a b : UnitDisc) : ℝ :=
  Real.arctanh (Complex.abs ((a - b) / (1 - conj a * b)))

-- Carathéodory pseudodistance: sup of Poincaré readings over Hol(Ω, Δ).
noncomputable def caratheodory (p q : Fin n → ℂ) : ℝ :=
  ⨆ f : HolomorphicTo Ω UnitDisc, poincare (f p) (f q)

-- Distance-decreasing property under a holomorphic map F : Ω → Ω'.
theorem caratheodory_decreasing
    (F : HolomorphicTo Ω Ω') (p q : Fin n → ℂ) :
    caratheodory Ω' (F p) (F q) ≤ caratheodory Ω p q := by
  sorry

-- Comparison c ≤ k.
theorem caratheodory_le_kobayashi (p q : Fin n → ℂ) :
    caratheodory Ω p q ≤ kobayashi Ω p q := by
  sorry

end Codex.SeveralComplexVariables.InvariantMetrics

The proof depends on names with no current Mathlib counterpart (the Poincaré metric instance, holomorphic-map spaces between domains, the sup/inf extremal definitions, normal-family compactness for the Carathéodory supremum, and Royden's integration theorem). Each is a candidate Mathlib contribution; until then this unit ships with lean_status: none.

Advanced results Master

The invariant-metric theory acquires its force through three structural facts beyond the distance-decreasing property: the integration theorem for the Kobayashi metric, the comparison with the Bergman metric, and the rigidity theorem identifying the two extremes on convex domains.

Royden's integration theorem. The infinitesimal Kobayashi metric ${\kappa }_{\Omega }(z;v)$ is upper semicontinuous on $\Omega \times {\mathbb{C}}^n$, and the Kobayashi pseudodistance is its integrated length:

$$k_{\Omega }(p,q)=\underset{\sigma }{\inf }{\int }_0^1{\kappa }_{\Omega }(\sigma (t);{\sigma }^{\prime }(t))dt,$$

the infimum over piecewise-$C^1$ paths $\sigma$ from $p$ to $q$. This was proved by Royden 1971 and converts the chain-of-discs definition into a length-space statement, putting $k_{\Omega }$ on the same footing as a Finsler metric. The analogous statement for the Carathéodory side is false in general — the integrated Carathéodory-Finsler length ${\hat{c}}_{\Omega }$ can strictly exceed $c_{\Omega }$ — and the discrepancy ${\hat{c}}_{\Omega }\ge c_{\Omega }$ with strict inequality on certain non-convex domains is one of the genuinely several-variable phenomena absent in dimension one.

The Lempert theorem. On a bounded convex domain $\Omega \subset {\mathbb{C}}^n$, the Carathéodory and Kobayashi distances coincide, $c_{\Omega }=k_{\Omega }$, and equal the integrated infinitesimal metric; moreover every pair of points lies on a complex geodesic — an analytic disc $\phi :\Delta \to \Omega$ that is a $k$-isometry onto its image and admits a holomorphic left inverse (Lempert 1981). The proof is a deep application of the theory of stationary discs and the homogeneity of the defining function; it has no one-variable shadow because in dimension one every domain becomes the disc after uniformization, where convexity is automatic. Lempert's theorem is the reason the ball computation of Exercise 6 extends to all convex domains, and it underpins the use of the Kobayashi metric in complex dynamics and the geometry of Teichmüller-type spaces.

Comparison with the Bergman metric. On a bounded domain all three invariants are positive-definite metrics, and there are universal comparison constants on strongly pseudoconvex domains: the boundary behaviour of ${\kappa }_{\Omega }$ and of the Bergman metric ${\beta }_{\Omega }$ both blow up like $(-\rho {)}^{-1}$ in the complex-normal direction and $(-\rho {)}^{-1/2}$ in complex-tangential directions, where $\rho$ is a defining function — the same anisotropic blow-up that drives the Bergman-kernel boundary asymptotics of 06.10.08. The Carathéodory metric is comparable from below; on strongly pseudoconvex domains all three are mutually boundedly equivalent, though they are not equal except on the ball.

Completeness and hyperbolicity. A domain $\Omega$ is Kobayashi hyperbolic when $k_{\Omega }$ separates points, and complete hyperbolic when $(\Omega ,k_{\Omega })$ is a complete metric space. Every bounded domain is hyperbolic (embed in a polydisc and use that the polydisc separates points). Bounded pseudoconvex domains are complete hyperbolic; on the Bergman side, bounded pseudoconvex domains are Bergman complete (Kobayashi; Ohsawa), the parallel completeness statement from 06.10.08. Hyperbolicity is the gateway to the big Picard theorem in several variables: a holomorphic map from a punctured disc into a complete hyperbolic domain extends across the puncture, because finite Kobayashi length forbids essential singularities.

Synthesis. The three invariant metrics are the universal upper and lower bounds, plus the canonical Hilbert-space bound, on every invariant reading of complex-analytic shape. The Carathéodory distance is the largest invariant distance built from bounded holomorphic functions and the Kobayashi distance is the smallest invariant distance dominating analytic discs; this is exactly the squeeze $c_{\Omega }\le k_{\Omega }$, and the Bergman metric from 06.10.08 sits inside it on bounded domains as the curvature of the reproducing kernel. Putting these together, Lempert's theorem identifies the two extremes on convex domains, which is the foundational reason the ball carries a single complex hyperbolic metric and the bidisc does not. Royden's integration theorem is dual to this rigidity: it makes the Kobayashi distance a genuine length structure, so that the distance-decreasing property generalises the one-variable Schwarz-Pick lemma into a Finsler-geometric statement. The central insight is that the Poincaré disc is the universal calibrating object, and every domain inherits its metric geometry by mapping holomorphically to and from the disc; completeness of these metrics on bounded pseudoconvex domains is what converts the local distance-decreasing inequality into the global hyperbolicity that governs holomorphic mappings, and this is the bridge from the function theory of 06.10.03 to the rigidity theory of automorphism groups.

Full proof set Master

Proposition (the Kobayashi pseudodistance is the largest among disc-dominated pseudodistances). Let $d$ be any pseudodistance on $\Omega$ such that every analytic disc $\phi \in \mathrm{Hol}(\Delta ,\Omega )$ is distance-decreasing, i.e. $d(\phi (a),\phi (b))\le \rho (a,b)$ for all $a,b\in \Delta$. Then $d\le k_{\Omega }$. Conversely $k_{\Omega }$ itself has this property, so it is the largest such pseudodistance.

Proof. Let $p,q\in \Omega$ and let ${\phi }_1,\dots ,{\phi }_m$ with parameters $a_i,b_i$ be any chain from $p$ to $q$. By hypothesis each disc satisfies $d({\phi }_i(a_i),{\phi }_i(b_i))\le \rho (a_i,b_i)$. Using ${\phi }_i(b_i)={\phi }_{i+1}(a_{i+1})$ and the triangle inequality for $d$,

$$d(p,q)\le \sum _{i=1}^{m}d({\phi }_i(a_i),{\phi }_i(b_i))\le \sum _{i=1}^m\rho (a_i,b_i).$$

Taking the infimum over chains gives $d(p,q)\le k_{\Omega }(p,q)$. For the converse, the single chain consisting of one disc $\phi$ shows $k_{\Omega }(\phi (a),\phi (b))\le \rho (a,b)$, so analytic discs are $k_{\Omega }$-distance-decreasing, and $k_{\Omega }$ is a pseudodistance because the chain functional satisfies the triangle inequality (concatenate chains). $\square$

Proposition (the Carathéodory pseudodistance is the smallest among function-dominating invariant pseudodistances). Among all pseudodistances $d$ on $\Omega$ such that $\rho (f(p),f(q))\le d(p,q)$ for every $f\in \mathrm{Hol}(\Omega ,\Delta )$, the Carathéodory pseudodistance $c_{\Omega }$ is the smallest.

Proof. If $d$ dominates every Poincaré pullback then $d(p,q)\ge {\sup }_f\rho (f(p),f(q))=c_{\Omega }(p,q)$, so $c_{\Omega }\le d$. And $c_{\Omega }$ itself dominates every pullback by its definition as the supremum. Hence $c_{\Omega }$ is the least element of the family. $\square$

Proposition (boundedness implies hyperbolicity). Every bounded domain $\Omega \subset {\mathbb{C}}^n$ is Kobayashi hyperbolic: $k_{\Omega }(p,q)>0$ for $p\ne q$.

Proof. After scaling and translating, assume $\Omega \subset {\Delta }^n$ (a polydisc), which is possible because $\Omega$ is bounded. The inclusion $\iota :\Omega \hookrightarrow {\Delta }^n$ is holomorphic, so distance-decreasing gives $k_{\Omega }(p,q)\ge k_{{\Delta }^n}(p,q)$. By the product formula (Exercise 3 generalised to $n$ factors) $k_{{\Delta }^n}(p,q)={\max }_j\rho (p_j,q_j)$, which is strictly positive when $p\ne q$ since some coordinate differs. Therefore $k_{\Omega }(p,q)>0$. $\square$

Proposition (comparison of infinitesimal metrics gives the distance comparison). If ${\gamma }_{\Omega }(z;v)\le {\kappa }_{\Omega }(z;v)$ for all $(z,v)$, then $c_{\Omega }\le k_{\Omega }$ as integrated distances on the Kobayashi side.

Proof. Royden's theorem expresses $k_{\Omega }(p,q)$ as the infimum of ${\int }_0^1{\kappa }_{\Omega }(\sigma ;{\sigma }^{\prime })dt$ over paths $\sigma$ from $p$ to $q$. The Carathéodory distance satisfies $c_{\Omega }(p,q)\le {\int }_0^1{\gamma }_{\Omega }(\sigma ;{\sigma }^{\prime })dt$ for any path, by integrating the infinitesimal distance-decreasing inequality along $\sigma$ and using that each Poincaré pullback is dominated by the $\gamma$-length. Combining with $\gamma \le \kappa$ pointwise and taking the path infimum on the right yields $c_{\Omega }(p,q)\le k_{\Omega }(p,q)$, recovering the comparison theorem through the infinitesimal route. $\square$

Connections Master

• Maximum modulus and the Schwarz lemma 06.01.12. The one-variable Schwarz-Pick lemma is the engine of the whole theory: it makes holomorphic self-maps of the disc Poincaré-distance-decreasing, and post/pre-composition with the disc lifts this to the Carathéodory and Kobayashi distance-decreasing properties on every domain. The invariant metrics are the $n$-variable extension of Schwarz-Pick.

• Bergman kernel and Bergman metric 06.10.08. The Bergman metric is the third invariant metric. Its biholomorphic invariance (the $\partial \bar{\partial }\log K$ construction) places it inside the $c\le k$ squeeze on bounded domains, and its boundary asymptotics on strongly pseudoconvex domains match those of the Kobayashi metric, giving mutual boundedness there.

• Pseudoconvexity and the Levi form 06.10.03. Completeness and hyperbolicity of the invariant metrics are governed by pseudoconvexity: bounded pseudoconvex domains are complete hyperbolic and Bergman complete, and the anisotropic boundary blow-up of the metrics is dictated by the Levi form of the defining function.

• Solution of the Levi problem 06.10.05. The function theory that makes the Carathéodory construction non-degenerate — the existence of bounded holomorphic functions separating points and directions — rests on the holomorphic-convexity theory settled by the Levi problem; on pseudoconvex domains the supply of holomorphic maps to the disc is rich enough to make $c_{\Omega }$ a genuine distance in good cases.

• Bergman kernel boundary asymptotics and the automorphism group 06.10.13 (pending). The invariant metrics are biholomorphism isometries, so the automorphism group $\mathrm{Aut}(\Omega )$ acts by isometries on $(\Omega ,k_{\Omega })$; Lempert's theorem and the metric rigidity feed directly into the structure of $\mathrm{Aut}(\Omega )$ as a Lie group and into the Fefferman mapping theorem.

Historical & philosophical context Master

Constantin Carathéodory introduced his pseudodistance in 1926 in Über das Schwarzsche Lemma bei analytischen Funktionen von zwei komplexen Veränderlichen ^{[Carathéodory 1926]} (Math. Ann. 97, 76-98), motivated explicitly by the problem of extending the Schwarz lemma to two complex variables. His construction — measure separation by the most expansive bounded holomorphic function into the disc — was the first biholomorphically invariant distance in several variables and showed that the rigidity of the one-variable Schwarz lemma survives in higher dimension, though now as an inequality between an upper and a lower invariant reading rather than a single sharp identity.

Shoshichi Kobayashi defined the complementary pseudodistance in 1967 in Invariant distances on complex manifolds and holomorphic mappings ^{[Kobayashi 1967]} (J. Math. Soc. Japan 19, 460-480; expanded in his 1976 Bull. AMS survey), building the largest invariant distance dominated by analytic discs. Kobayashi's chain-of-discs construction made the distance-decreasing property automatic and connected the metric to the value-distribution theory of Nevanlinna and the big Picard theorem, defining hyperbolic complex manifolds as those on which his distance separates points. Hugh Royden's 1971 theorem ^{[Royden 1971]} (Springer LNM 185) showed the Kobayashi distance is the integrated length of an infinitesimal Finsler metric, recasting the chain definition as length-geometry.

The deepest structural result is László Lempert's 1981 theorem ^{[Lempert 1981]} (Bull. Soc. Math. France 109, 427-474) that the Carathéodory and Kobayashi distances coincide on convex domains via complex geodesics, resolving the question of when the upper and lower invariant readings agree and establishing the role of stationary analytic discs. The encyclopaedic modern treatment is Jarnicki and Pflug's Invariant Distances and Metrics in Complex Analysis ^{[Jarnicki-Pflug]}, which catalogues the product formulas, completeness criteria, and the gap between the integrated and non-integrated Carathéodory distances.

Bibliography Master

@article{Caratheodory1926,
  author  = {Carath{\'e}odory, Constantin},
  title   = {{\"U}ber das {Schwarzsche} {Lemma} bei analytischen {Funktionen} von zwei komplexen {Ver{\"a}nderlichen}},
  journal = {Math. Ann.},
  volume  = {97},
  year    = {1926},
  pages   = {76--98}
}

@article{Kobayashi1967,
  author  = {Kobayashi, Shoshichi},
  title   = {Invariant distances on complex manifolds and holomorphic mappings},
  journal = {J. Math. Soc. Japan},
  volume  = {19},
  year    = {1967},
  pages   = {460--480}
}

@article{Kobayashi1976,
  author  = {Kobayashi, Shoshichi},
  title   = {Intrinsic distances, measures and geometric function theory},
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  volume  = {82},
  year    = {1976},
  pages   = {357--416}
}

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  author    = {Kobayashi, Shoshichi},
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  publisher = {Springer},
  year      = {1998}
}

@incollection{Royden1971,
  author    = {Royden, Halsey L.},
  title     = {Remarks on the {Kobayashi} metric},
  booktitle = {Several Complex Variables II (Maryland 1970)},
  series    = {Lecture Notes in Mathematics},
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  publisher = {Springer},
  year      = {1971},
  pages     = {125--137}
}

@article{Lempert1981,
  author  = {Lempert, L{\'a}szl{\'o}},
  title   = {La m{\'e}trique de {Kobayashi} et la repr{\'e}sentation des domaines sur la boule},
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  year    = {1981},
  pages   = {427--474}
}

@book{KrantzSCV,
  author    = {Krantz, Steven G.},
  title     = {Function Theory of Several Complex Variables},
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}

@book{JarnickiPflug,
  author    = {Jarnicki, Marek and Pflug, Peter},
  title     = {Invariant Distances and Metrics in Complex Analysis},
  edition   = {2},
  series    = {de Gruyter Expositions in Mathematics},
  volume    = {9},
  publisher = {Walter de Gruyter},
  year      = {2013}
}