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arXiv · 1512.03858

Complex length of short curves and Minimal Fibration in hyperbolic $3$-Manifolds fibering over the circle

Zheng Huang·Biao Wang
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Abstract

We investigate the maximal solid tubes around short simple geodesics in hyperbolic three-manifolds and how complex length of curves relate to closed, incompressible, least area minimal surfaces. As applications, we prove, there are some closed hyperbolic three-manifolds fibering over the circle which are not foliated by closed incompressible minimal surfaces diffeomorphic 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 we find explicit examples where these conditions are satisfied via computer programs.

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Zheng Huang, Biao Wang. 2018-11-28. Complex length of short curves and Minimal Fibration in hyperbolic $3$-Manifolds fibering over the circle. https://doi.org/10.1112/plms.12216

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