Carathéodory metric

In mathematics, the Carathéodory metric is a metric defined on the open unit ball of a complex Banach space that has many similar properties to the Poincaré metric of hyperbolic geometry. It is named after the Greek mathematician Constantin Carathéodory.

Definition

Let (X, || ||) be a complex Banach space and let B be the open unit ball in X. Let Δ denote the open unit disc in the complex plane C, thought of as the Poincaré disc model for 2-dimensional real/1-dimensional complex hyperbolic geometry. Let the Poincaré metric ρ on Δ be given by

${\displaystyle \rho (a,b)=\tanh ^{-1}{\frac {|a-b|}{|1-{\bar {a}}b|}}}$

(thus fixing the curvature to be −4). Then the Carathéodory metric d on B is defined by

${\displaystyle d(x,y)=\sup\{\rho (f(x),f(y))|f:B\to \Delta {\mbox{ is holomorphic}}\}.}$

What it means for a function on a Banach space to be holomorphic is defined in the article on Infinite dimensional holomorphy.

Properties

• For any point x in B,

${\displaystyle d(0,x)=\rho (0,\|x\|).}$

• d can also be given by the following formula, which Carathéodory attributed to Erhard Schmidt:

${\displaystyle d(x,y)=\sup \left\{\left.2\tanh ^{-1}\left\|{\frac {f(x)-f(y)}{2}}\right\|\right|f:B\to \Delta {\mbox{ is holomorphic}}\right\}}$

• For all a and b in B,

${\displaystyle \|a-b\|\leq 2\tanh {\frac {d(a,b)}{2}},\qquad \qquad (1)}$

with equality if and only if either a = b or there exists a bounded linear functional ℓ ∈ X^∗ such that ||ℓ|| = 1, ℓ(a + b) = 0 and

${\displaystyle \rho (\ell (a),\ell (b))=d(a,b).}$

Moreover, any ℓ satisfying these three conditions has |ℓ(a − b)| = ||a − b||.

• Also, there is equality in (1) if ||a|| = ||b|| and ||a − b|| = ||a|| + ||b||. One way to do this is to take b = −a.
• If there exists a unit vector u in X that is not an extreme point of the closed unit ball in X, then there exist points a and b in B such that there is equality in (1) but b ≠ ±a.

Carathéodory length of a tangent vector

There is an associated notion of Carathéodory length for tangent vectors to the ball B. Let x be a point of B and let v be a tangent vector to B at x; since B is the open unit ball in the vector space X, the tangent space T_xB can be identified with X in a natural way, and v can be thought of as an element of X. Then the Carathéodory length of v at x, denoted α(x, v), is defined by

${\displaystyle \alpha (x,v)=\sup {\big \{}|\mathrm {D} f(x)v|{\big |}f:B\to \Delta {\mbox{ is holomorphic}}{\big \}}.}$

One can show that α(x, v) ≥ ||v||, with equality when x = 0.

See also

• Earle–Hamilton fixed point theorem

References

• Earle, Clifford J. and Harris, Lawrence A. and Hubbard, John H. and Mitra, Sudeb (2003). "Schwarz's lemma and the Kobayashi and Carathéodory pseudometrics on complex Banach manifolds". In Komori, Y.; Markovic, V.; Series, C. (eds.). Kleinian groups and hyperbolic 3-manifolds (Warwick, 2001). London Math. Soc. Lecture Note Ser. 299. Cambridge: Cambridge Univ. Press. pp. 363–384.CS1 maint: multiple names: authors list (link)