We’re going to get to quasi-conformal maps soon, but first, we’re going to want to build some new objects on our Riemann surfaces. Differential forms are a fairly standard thing, and asserting that they’re holomorphic isn’t exactly a revolution. On Riemann surfaces, of course, there’s holomorphic 1-forms only, by dimension reasons. So now, we’ll talk about quadratic forms, and a few of the things we can do with them.
So we start with a definition, as we so often do:
Definition: A quadratic differential is a section of , the tensor square of the holomorphic cotangent bundle. It can be written locally as where is holomorphic.
These, in some sense, generalize holomorphic differentials (also called abelian, due to Abel’s study of their integrals) because the square of holomorphic differentials are quadratic differentials (in fact, this route leads us to the Veronese map, but that’s a topic for another day).
So, a quick calculation using Riemann-Roch tells us that the space of quadratic differentials is dimensional on a compact Riemann surface. This is not a coincidence! Some other time, we’ll connect the quadratic differentials more directly to deformation theory and show that this space is naturally dual to the tangent space to moduli. For now, though, let’s see what we can get more immediately.
I dodged talking about foliations with respect to pseudo-Anosov maps last time. This time, though, I can talk about them briefly with respect to quadratic differentials. Let be the zero locus of a quadratic differential . Then, away from , our quadratic differential is a quadratic function on each tangent space, we get line fields given by the vectors on which is positive and negative, called respectively horizontal and vertical.
In fact, these lines are horizontal and vertical lines in a distinguished coordinate: let be a nonzero of , then locally near , we have that is nowhere zero, then we can take any coordinate at and take a branch of the square root to write on some neighborhood containing . Then, we can build a coordinate, evaluated at a point near , by
This coordinate is unique, up to translation and multiplication by , such that . And it’s this coordinate that makes the foliation lines parallel to the real and imaginary axes.
We’ll finish off by noting that the singular metric given by is of finite area, so we actually get a norm on the space of quadratic differentials (which let’s just establish the notation for, though it’s often denoted by ) given by .