I'm trying to solve a PDE on compact Riemann surfaces of genus g > 1. Since these can be obtained as quotients of the upper half plane $\mathbb{H}_2$ by some Fuchsian group $\Gamma$, I suppose it's best to solve things on $\mathbb{H}_2$ first and impose invariance under $\Gamma$ later.

On $\mathbb{H}_2$, the equation in question is $\frac{\partial}{\partial \bar{z}} X = -\frac{\imath q}{4Im z} X$. Here $q \in \mathbb{R}$ is constant and $Im z$ is the imaginary part of $z$. On $\mathbb{H}_2$, the solution is then $X = (Im z)^{-q/2} f(z)$ for any holomorphic function $f$.

To take the quotient, this should be invariant under the action of $\Gamma$. I'd like to consider a simple example. Donaldson's book on Riemann Surfaces states that the Klein Curve is a compact Riemann surface of genus $3$ with $\Gamma = \{M \in PSL(2,Z)| M \equiv 1 \mod 7\}$. I wasn't able to find $\Gamma$ for the Bolza surface, which might be an even easier example.

Invariance under $z \mapsto z+7 \in \Gamma$ means that (locally) $f(z) = \sum_k a_k e^{2\pi \imath k z/7}$. If one considers next the action of $z \mapsto \frac{z}{7z+1}$, one obtains $(Im z)^{-q/2} \sum_k a_k e^{\frac{2\pi \imath k z}{7(7z+1)}} [ (7 Re z+1)^2 + (7Im z)^2]$. Now, since the term in angular brackets is not holomorphic, this cannot be invariant and thus it seems that the only solution is actually $X = constant$.


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