The (measurable) Riemann mapping theorem

Question

The Riemann mapping theorem says that a strict, nonempty open subset of the complex plane is conformally equivalent to the unit disk.

The measurable Riemann mapping theorem asserts the existence and uniqueness of a quasiconformal homeomorphism $f$ satisfying the Beltrami equation: $$\frac{\partial f}{\partial \overline{z}} = \mu(z)\frac{\partial f}{\partial z} $$ for given $\mu$ with $ \lVert\mu\rVert_{\infty}<1$. What (if anything) do these two statements have to do with each other? Wikipedia points out in the link above that the latter isn't a direct generalization of the former, although there does seems to be a proof of Riemann mapping theorem from the measurable RMT.

Answer 1

You misstated Riemann's (original) theorem: a crucial assumption is that your open subset is simply connected.

Both theorems can be considered as classification theorems of Riemann surfaces. The Riemann original theorem says that every simply connected domain in the sphere, whose complement contains at least 2 points is conformally equivalent to the unit disk.

"Measurable Riemann theorem" says that a sphere equipped with any Riemannian metric, subject to certain condition (that the Beltrami coefficient $\mu$ has norm <1) is conformally equivalent to the Riemann sphere.

It has simple corollaries that a plane or a disk equipped with a Riemannian metric satisfying the same condition are conformally equivalent to the plane and disk respectively with the standard metric. (A disk with arbitrary Riemannian metric is a generalization of a simply connected domain in the plane with the usual metric).

The old, classical name of the "Measurable Riemann theorem" was "Existence and uniqueness theorem for Beltrami equation", or it was called simply by the name of an author (Korn and Lichtenstein, or Morrey or Boyarski, depending on exact conditions, and the taste of the person who refers). Boyarski's contribution is the very important fact that properly normalized $f$ depends on $\mu$ analytically.

The modern name comes from the paper of Ahlfors and Bers, Riemann's mapping theorem for variable metrics, Ann. Math., 72 2 (1960), 385-404, where they restated the result of Boyarski in the spirit that I outlined above, and emphasized this analytic dependence on $\mu$. Besides $\|\mu\|_\infty<1$, no condition on $\mu$ is imposed except that it is Lebesgue measurable, and the word "measurable" in the name of the theorem comes from this fact.