Fixed points on Riemann surface

Question

It is well known theorem that for a conformal mapping $\phi$ from a bounded and planar domain $\Omega$ to itself has three fixed points , then it must be identity mapping. However, I cannot find a elementary proof for it?

Any comments and reference will be appreciated

Answer 1

This theorem is an immediate consequence of a result by B. Maskit, which states that one may associate with $\Omega$ another domain $\Omega'$, conformally equivalent to $\Omega$, such that all conformal self-maps of $\Omega'$ are Möbius transformations. For a proof of Maskit's theorem, I suggest you look at the following papers :

1. Maskit, Bernard The conformal group of a plane domain. Amer. J. Math. 90 1968 718–722. 30.45

or for a more elementary proof :

1. Peschl, Ernst; Lehtinen, Matti A conformal self-map which fixes three points is the identity. Ann. Acad. Sci. Fenn. Ser. A I Math. 4 (1979), no. 1, 85–86.