TITLE: Holomorphic deformation of complex structure on the real plane
QUESTION [8 upvotes]: It is known that each complex structures on $\mathbb{R}^2$ is biholomorphic to either $\mathbb{C}$ or the open unit disk $\Delta$.
One can continuously deform one complex structure to the other as is for example done in Winkelmann - Deformations of Riemann surfaces (page 3).
My question is 

Can this deformation be taken to be holomorphic on the deformation parameter? That is, does there exist a non-trivial complex analytic family $M \to D$ where $D \subset \mathbb{C}$ is a small disk, the central fiber is biholomorphic to $\mathbb{C}$, and the generic fiber is biholomorphic to $\Delta$?

Note that all the theorems that assure complex analytic triviality of deformations when $H^1(X,TX)$ vanishes, use the hypothesis that $X$ is compact.

REPLY [7 votes]: The answer is no. This follows from the so-called $\lambda$-lemma of Sullivan, Mane Sad and Lyubich: Let $D$ be a disk, $C$ the complex plane and $A$ any set in C.
Let $f:D\times A\to C$ be a function with the following properties:

$\lambda\mapsto f(\lambda,z)$ is holomorphic for every $z\in A$.
$z\mapsto f(\lambda,z)$ is injective for every $\lambda\in D$,
$z\mapsto f(0,z)=z$ for every $z\in A$.

Then for every $\lambda\in D$ the map $z\mapsto f(\lambda,z)$ is quasisymmetric (=quasiconformal if $A$ is open)
Since the plane is not conformally equivalent to a disk, this implies that the
answer to your question is negative.
Ref. MR0732343 
Mañé, R.; Sad, P.; Sullivan, D.,
On the dynamics of rational maps,
Ann. Sci. École Norm. Sup. (4) 16 (1983), no. 2, 193–217.