Classification of holomorphic disc bundles 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?
 A: There's a paper of H. L. Royden  
"Holomorphic fiber bundles with hyperbolic fiber", Proc. AMS, Volume 43, Number 2, April 1974 
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.
A: If I don't misunderstand the question, I don't  believe that any kind of such classification can exists in the case when the normal bundle of the surface has positive degree. This moduli is infinite-dimensional. 
Even in the case of negative degree there are problems. If we take a disk bundle over $\mathbb CP^1$ of degree -1, and contract $\mathbb CP^1$ we will get in particular plenty of complex balls in $\mathbb C^2$. Can one classify them up to biholomorphism? 
