Conformally-invariant metric

on a Riemann surface $ R $

A rule that associates with each local parameter $ z $, mapping a parameter neighbourhood $ U \subset R $ into the closed complex plane $ \overline{\mathbf C}\; $( $ z : U \rightarrow \overline{\mathbf C}\; $), a real-valued function

$$ \rho _ {z} : z ( U) \rightarrow \ [ 0 , + \infty ] $$

such that for all local parameters $ z _ {1} : U \rightarrow \overline{\mathbf C}\; $ and $ z _ {2} : U _ {2} \rightarrow \overline{\mathbf C}\; $ for which the intersection $ U _ {1} \cap U _ {2} $ is not empty, the following relation holds:

$$ \frac{\rho _ {z _ {2} } ( z _ {2} ( p) ) }{\rho _ {z _ {1} } ( z _ {1} ( p) ) } = \ \left | \frac{d z _ {1} ( p) }{d z _ {2} ( p) } \ \right | \ \ ( \forall p \in U _ {1} \cap U _ {2} ) , $$

where $ z ( U) $ is the image of $ U $ in $ \overline{\mathbf C}\; $ under $ z $. A conformally-invariant metric is often denoted by the symbol $ \rho ( z) | d z | $, to which the indicated invariance with respect to the choice of the local parameter $ z $ is attributed.

Every linear differential $ \lambda ( z) d z $( or quadratic differential $ Q ( z) d z ^ {2} $) induces a conformally-invariant metric, $ | \lambda ( z) | \cdot | d z | $( or $ | Q ( z) | ^ {1/2} | d z | $). The notion of a conformally-invariant metric, being a very general form of defining conformal invariants, enables one to introduce that of the length of curves on $ R $ as well as the notion of the extremal length and the modulus of families of curves (see Extremal metric, method of the, and also [1]). The definition of a conformally-invariant metric can be carried over to Riemann varieties of arbitrary dimension.

References

[1]	J.J. Jenkins, "Univalent functions and conformal mapping" , Springer (1958)
[2]	M. Schiffer, D.C. Spencer, "Functionals of finite Riemann surfaces" , Princeton Univ. Press (1954)
[3a]	L.V. Ahlfors, "The complex analytic structure of the space of closed Riemann surfaces" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 45–66
[3b]	L.V. Ahlfors, "On quasiconformal mappings" J. d'Anal. Math. , 3 (1954) pp. 1–58
[3c]	L.V. Ahlfors, "Correction to "On quasiconformal mappings" " J. d'Anal. Math. , 3 (1954) pp. 207–208
[3d]	L. Bers, "Quasi-conformal mappings and Teichmüller's theorem" R. Nevanlinna (ed.) et al. (ed.) , Analytic functions , Princeton Univ. Press (1960) pp. 89–119
[3e]	L. Bers, "Spaces of Riemann surfaces" J.A. Todd (ed.) , Proc. Internat. Congress Mathematicians (Edinburgh, 1958) , Cambridge Univ. Press (1960) pp. 349–361
[3f]	L. Bers, "Simultaneous uniformization" Bull. Amer. Math. Soc. , 66 (1960) pp. 94–97
[3g]	L. Bers, "Holomorphic differentials as functions of moduli" Bull. Amer. Math. Soc. , 67 (1961) pp. 206–210

References

[a1]	L.V. Ahlfors, "Lectures on quasiconformal mappings" , v. Nostrand (1966)