Ok, time to get back to Riemann surfaces! We’ve been all about hyperbolic surfaces, and so first let’s compare the two of them: every oriented hyperbolic surface is a Riemann surface, and the conformal class of a complex structure contains a unique hyperbolic metric.
This fairly straightforward result (the first part is the fact that orientation preserving isometries of the hyperbolic plane are biholomorphic and so define a complex structure, and the second is from the uniformization theorem) is why the moduli of curves (complex structures on a surface) is just the moduli of marked hyperbolic surfaces (Teichmüller space) after getting rid of the marking (quotient by the mapping class group).
So now, we’ll start looking at the geometry of Teichmüller space. One of the main tools there is the notion of a quasiconformal map.
Definition: Let and be Riemann surfaces. Let . Then a map is -quasiconformal if is a homeomorphism with continuous differential of maximal rank outside of a finite set of points, and we assume that outside of , with and .
What this means is that away from , the differential takes a circle to an ellipse in the tangent spaces, and the ratio of the lengths of the axes is bounded by .
So what can we do with this? Next time, we’ll really get into a big application of quasi-conformal maps and quadratic differentials, but now, we’ll just look at a couple of basic properties.
We can use quadratic differentials to build some special quasi-conformal maps. Let be a quadratic differential. It gives us a singular Euclidean metric, and we can scale it. In the horizontal direction, we scale by and in the vertical direction by , which gives us a quasiconformal map with constant , and we call these maps Teichmüller maps with initial differential and stretch factor , and they’re of constant area.
This lets us compute:
Theorem: Let be a quasiconformal map which is homotopic to a Teichmuller mapping with initial differential and stretch factor . Then where is the derivative of in the horizontal direction and the norm is with respect to the singular metric from .
This can be used to classify annuli. An annulus is where for all . Then for , the smallest quasiconformal dilation is .
This, then, implies that if and only if .
We’ll close with a theorem of Wolpert:
Let and be hyperbolic surfaces and let be a -quasiconformal map. Then for any simple closed curve , we have , where is the length as we discussed before. So that gives a nice interpretation of the in quasiconformal maps: it’s a bound on how much simple closed curves change their length.