Complex length of short curves and minimal fibrations of hyperbolic three-manifolds fibering over the circle. (English) Zbl 1418.53066
Summary: 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.
MSC:
| 53C42 | Differential geometry of immersions (minimal, prescribed curvature, tight, etc.) |
| 53C22 | Geodesics in global differential geometry |
| 53A10 | Minimal surfaces in differential geometry, surfaces with prescribed mean curvature |
| 30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |
| 57M05 | Fundamental group, presentations, free differential calculus |
Keywords:
maximal solid tubes; simple closed geodesics; incompressible minimal surfaces; quasi-Fuchsian manifoldsReferences:
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