Obstructions to minimal fibrations of hyperbolic 3-manifolds

Through the work of Agol and Wise, we know that all closed hyperbolic 3-manifolds are finitely covered by a surface bundle over the circle. Thus the geometry of these bundles indicates the geometry of general hyperbolic 3-manifolds. But there are still many open problems about these bundles. The main result that I'll discuss shows that there are hyperbolic 3-manifolds that fiber over the circle but that do not admit fibrations by minimal surfaces. These manifolds do not admit fibrations by surfaces that are even approximately minimal. I will also discuss some open problems about shortest geodesics.