Calculation of dimension of holomorphic quadratic differentials as in Gardiners book

Question

In Frederick Gardiner's book Teichmuller Theory and Quadratic Differentials, P.27-28, Chapter 1 ) that dimension of $dim_RQD(X) = 6g-6+3m+2n $ ( by using Riemann-Roch theorem ). Now for open annulus $A$, $ g=0, m=2, n=0 $, we get $ dim_RQD(X)=0 $ ! I am a bit puzzled why it is zero ! (Should I define the genus of an open annulus to be zero ?)

For q.diffs $q$ on the annulus $A$, should we look at $ q=\phi(z)dz^2 $ when $\phi$ is a function on the annulus embedded in complex plane or should we lift it to upper half plane and consider the $ \phi(z) $ with $ \phi(z) = \phi(\gamma(z)) ({\gamma'(z)})^2$ for all $\gamma \in Deck(H/A) $ ? I guess the second approach makes more sense because it respects the hyperbolic geometric structure on $A$ as well ?

Answer 1

This mostly counts as a comment since I don't have enough reputation to make one. Gardiner mentioned right before the statement of the dimension counting that

Along a border arc $\alpha$ in the border of $R$ we require that $\varphi(z_1)$ be real if $z_1$ is a local coordinate taking real values along $\alpha$.

If you try a local $\log$ chart along a small neighborhood of a small arc on the boundary $\partial A$ of the open annulus $A$, then $\varphi$ is greatly restricted by the rule above and you can argue that $\varphi=0$.

Plus, it's a fairly reasonable assumption since the boundary component corresponding to a finite modulus end should really behave like $\partial\mathbb{H}$. Without the extra requirement, you are likely to wind up with an infinite-dimensional space of quadratic differentials.

Still I'm not sure how to come up with the precise terms in the above quote. In comparison, we only allow simple poles at punctures so as to get finite total $\varphi$-area.