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Carathéodory metric

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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

ρ ( a , b ) = tanh 1 | a b | | 1 a ¯ b | {\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

d ( x , y ) = sup { ρ ( f ( x ) , f ( y ) ) | f : B Δ  is holomorphic } . {\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,
d ( 0 , x ) = ρ ( 0 , x ) . {\displaystyle d(0,x)=\rho (0,\|x\|).}
  • d can also be given by the following formula, which Carathéodory attributed to Erhard Schmidt:
d ( x , y ) = sup { 2 tanh 1 f ( x ) f ( y ) 2 | f : B Δ  is holomorphic } {\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,
a b 2 tanh d ( a , b ) 2 , ( 1 ) {\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
ρ ( ( a ) , ( b ) ) = d ( a , b ) . {\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 TxB 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(xv), is defined by

α ( x , v ) = sup { | D f ( x ) v | | f : B Δ  is holomorphic } . {\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(xv) ≥ ||v||, with equality when x = 0.

See also

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. ISBN 9780521540131.{{cite book}}: CS1 maint: multiple names: authors list (link)
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