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.

Definition: A quadratic differential $q$ is a section of $T^*X\otimes T^*X$, the tensor square of the holomorphic cotangent bundle.  It can be written locally as $q=f(z)dz^2$ where $f(z)$ 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 $3g-3$ 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 $\Sigma$ be the zero locus of a quadratic differential $q$.  Then, away from $\Sigma$, our quadratic differential is a quadratic function on each tangent space, we get line fields given by the vectors on which $q$ is positive and negative, called respectively horizontal and vertical.

In fact, these lines are horizontal and vertical lines in a distinguished coordinate: let $x$ be a nonzero of $q$, then locally near $x$, we have that $q$ is nowhere zero, then we can take any coordinate $u$ at $x$ and take a branch of the square root to write $q=g^2du^2$ on some neighborhood containing $x$.  Then, we can build a coordinate, evaluated at a point $y$ near $x$, by

$z(y)=\int_x^y g(u)du$

This coordinate is unique, up to translation and multiplication by $-1$, such that $q=dz^2$.  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 $q$ is of finite area, so we actually get a norm on the space of quadratic differentials (which let’s just establish the notation $\mathscr{Q}(X)$ for, though it’s often denoted by $H^0(X,2K)$) given by $||q||=\mathrm{area}(q)=\int_X |q|$.