show/hide this revision's text 2 Correction: derivative can't always be 1.

There is a very good theory of the boundary behavior for Riemann maps, the Caratheodory theory of ends. Riemann maps for an open set extend continuously to the boundary of a disk provided the frontier is locally connected.

I assume your definition is in terms of the Riemann maps that fix the north or south pole and have derivative the identity (1) a positive real number at those points. If so, then Riemann maps depend continuously in the $L^\infty$ metric on Jordan curves as you have described, so this gives a metric.

Is the metric complete? A sequence of the pair (lower hemisphere map, upper hemisphere map) that is Cauchy in the uniform ($L^\infty$) topology converges to a pair of continuous maps that agree on the equator, so the glued maps give a continuous image of the sphere that is a homeomorphism at least in the complement of the equator. If it is not injective, the preimages of points would need to be intervals, otherwise the topology would be destroyed. But that's impossible. If you push forward the measure $ds$ on the equator by a Riemann map, it never has atoms. It is the same as hitting measure for Brownian motion in the image: if you start at the north pole and follow a Brownian path, where does it first arrive at the boundary? This is always a diffuse measure.

So, you're right: it's a complete metric on the set of Jordan curves you described.

Note. Any quasisymmetric map (I won't define it here, but Holder is sufficient) from the circle to the circle arises from a pair of Riemann maps for a Jordan curve, although not necessarily in the annulus you described, and the Jordan curve in this case is known as a quasi-circle. However, there is no known good characterization of which gluing maps from the circle to the circle give Jordan curves in general. Continuity is not sufficient: there are counterexamples using things that are locally (for example) the graph of $\sin(1/x)$ plus the interval $[-1,1]$ on the $y$-axis, where the gluing map for the Riemann maps to the two sides extends continuously across the discontinuity of the graph. Better characterizations of what gluing maps give what topology and geometry are very hard, but of great interest in complex dynamics and some other subjects.
show/hide this revision's text 1

There is a very good theory of the boundary behavior for Riemann maps, the Caratheodory theory of ends. Riemann maps for an open set extend continuously to the boundary of a disk provided the frontier is locally connected.

I assume your definition is in terms of the Riemann maps that fix the north or south pole and have derivative the identity (1) at those points. If so, then Riemann maps depend continuously in the $L^\infty$ metric on Jordan curves as you have described, so this gives a metric.

Is the metric complete? A sequence of the pair (lower hemisphere map, upper hemisphere map) that is Cauchy in the uniform ($L^\infty$) topology converges to a pair of continuous maps that agree on the equator, so the glued maps give a continuous image of the sphere that is a homeomorphism at least in the complement of the equator. If it is not injective, the preimages of points would need to be intervals, otherwise the topology would be destroyed. But that's impossible. If you push forward the measure $ds$ on the equator by a Riemann map, it never has atoms. It is the same as hitting measure for Brownian motion in the image: if you start at the north pole and follow a Brownian path, where does it first arrive at the boundary? This is always a diffuse measure.

So, you're right: it's a complete metric on the set of Jordan curves you described.

Note. Any quasisymmetric map (I won't define it here, but Holder is sufficient) from the circle to the circle arises from a pair of Riemann maps for a Jordan curve, although not necessarily in the annulus you described, and the Jordan curve in this case is known as a quasi-circle. However, there is no known good characterization of which gluing maps from the circle to the circle give Jordan curves in general. Continuity is not sufficient: there are counterexamples using things that are locally (for example) the graph of $\sin(1/x)$ plus the interval $[-1,1]$ on the $y$-axis, where the gluing map for the Riemann maps to the two sides extends continuously across the discontinuity of the graph. Better characterizations of what gluing maps give what topology and geometry are very hard, but of great interest in complex dynamics and some other subjects.