Suppose we have a surface bundle $M$ over the unit circle $S^1$ and $M$ is assigned with Riemannian metric $g$. The projection map $\pi: M\to S^1$ now can be homotopic to a harmonic map $\phi: M\to S^1$, which is unique up to a constant function from $M$ to $S^1$. So, my question is: in general, is $\phi$ still a fiber bundle?
(The existence of $\phi$ can be shown as follows. We can find a smooth representative $f$ in the homotopy class of $\pi$, and locally we can lift $S^1$ to $\mathbb{R}$ and define $\Delta_g f$. We solve the equation $\Delta_g h=\Delta_g f$ and put $\phi=fe^{-2\pi ih}$)
If not, can we give some reasonable condition on the topology of $M$ or on the metric $g$ that will guarantee $\phi$ is a fiber bundle? In particular, in the case that $M$ is a hyperbolic 3-manifold (and $g$ is the hyperbolic metric), is $\phi$ a fiber bundle?