Disk bundles over surfaces - extending automorphisms of the boundary over the whole space Let $\Sigma$ be a closed connected orientable surface of genus $g \geq 2$.  I have been told that every diffeomorphism $\phi: \Sigma \times S^1 \to \Sigma \times S^1$ extends to a diffeomorphism $\Phi : \Sigma \times D^2 \to \Sigma \times D^2$.  I would love to know a reference for this fact and I would love to know if a similar thing happens for other orientable disk bundles over $\Sigma$.  
Namely, I want to know if given an orientable $D^2$-bundle $X^4$ over $\Sigma$ and a diffeomorphism $\phi : \partial X \to \partial X$ if there is an extension to a diffeomorphism $\Phi : X \to X$?  
I know that for the trivial disk bundle over a torus we do not have the above extension property.  What about for other orientable disk bundles over the torus?
 A: The answer to your first question is yes.  Suppose that $M$ is a closed Seifert fibered space with (a) vertical tori and (b) hyperbolic base orbifold.  Suppose that $f \colon M \to M$ is a self-diffeomorphism.  Then $f$ is isotopic to a fiber-preserving diffeomorphism.  See the proof of Proposition 2.4 in Hatcher's notes on three-manifolds. 
What you want does not quite follow from the statement of Proposition 2.4.  But the statement there can be improved: the diffeomorphism in the conclusion is in fact isotopic to the diffeomorphism in the hypothesis. 
Note that the three-torus has many non-isotopic diffeomorphisms.  This is why we want to exclude the possibility of the base surface being a two-torus.  It is not clear to me if there are  other (orientable) circle bundles over the torus with this property.   I'll guess that the answer is "no".  The natural candidates are Nil manifolds, and there the fibering is unique up to isotopy.  This could be made precise by examining the proof of Proposition 2.3 in Hatcher's notes and seeing where isomorphism can be replaced by isotopy.
