Complex length of short curves and Minimal Fibrations of hyperbolic three‐Manifolds fibering over the circle

We investigate the maximal solid tubes around short simple closed geodesics in hyperbolic three-manifolds and how the complex length of curves relates to closed least area incompressible minimal surfaces. As applications, we prove the existence of closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces isotopic to the fiber. We also show the existence of quasi-Fuchsian manifolds containing arbitrarily many embedded closed incompressible minimal surfaces. Our strategy is to prove main theorems under natural geometric conditions on the complex length of closed curves on a fibered hyperbolic three-manifold, then by computer programs, we find explicit examples where these conditions are satisfied.