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

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.

final version, to appear at the Proceeding of the London Math. Soc

References in corpus (2)

• Problems around 3-manifolds
• Stability of Catenoids and Helicoids in Hyperbolic Space

Cited by in corpus (3)

• Minimal Surfaces in Hyperbolic 3-manifolds
• Harmonic Forms, Minimal Surfaces and Norms on Cohomology of Hyperbolic $3$-Manifolds
• Minimal surfaces near short geodesics in hyperbolic $3$-manifolds