I've had difficulty finding sources which treat the classification of holomorphic disc bundles over (compact and noncompact) Riemann surfaces. Note that by "bundle", I mean a holomorphic fiber bundle, which means it is locally holomorphically trivial. 


I'm really just looking for information about holomorphic disc bundles, but if you want a specific question: 


What is the classification of holomorphic disc bundles over a Riemann surface?



Classification of holomorphic disc bundles
There's a paper of H. L. Royden 


["Holomorphic fiber bundles with hyperbolic fiber", Proc. AMS, Volume 43, Number 2, April 1974](http://www.ams.org/journals/proc/1974-043-02/S0002-9939-1974-0338465-0/S0002-9939-1974-0338465-0.pdf) 


which proves


"Theorem: The holomorphic fiber bundles with hyperbolic fiber 
M and base B are in natural one-to-one correspondence with the homomor- 
phisms of the fundamental group of B into the group of biholomorphic 
automorphisms of M onto itself. "


In other words, it appears that holomorphic disc bundles are naturally flat.


**Note**: the first projection $p\_1: \mathbb{B}^2\to \mathbb{D}$ of the real 4-ball to the disc is not a holomorphic bundle of discs, because it is not locally trivial. To see why, suppose we have an isomorphism $f:p\_1^{-1}(D)\to D\times \mathbb{D}$ of bundles over a sufficently small disc $D$ of radius $r<1$ around zero. Write $f(x,y) = (x, f\_2(x,y))$. 


Now choose $y\_0\in p\_1^{-1}(0)$ with $\sqrt{1-r^2} <|y\_0|< 1$, so that it is sent via the second projection $p\_2:\mathbb{B}^2\to\mathbb{D}$ outside the disc of radius $\sqrt{1-r^2}$. 


By assumption, $p\_1$ has a section $s$ over $D$ passing through $y\_0$ -- this is just a level set of $f\_2$. Composing, we obtain $ h=p\_2\circ s : D\to \mathbb{D}$. I claim $h$ fails the maximum modulus principle for holomorphic functions. Why? Any circle of sufficiently large radius in $D$ must be sent by $h$ inside a disc of radius strictly less than $|y\_0|$ around $0$. Meanwhile, the center of the disc $D$ is sent to $p\_2(y\_0)$, which has norm equal to $|y\_0|$. This goes against the maximum principle or the Gauss mean value theorem, if you like.


