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

Abstract

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.

Mean curvature flow in homology and foliations of hyperbolic $3$-manifolds

“…One might wonder if this constant mean curvature foliation can be extended globally. This is known to be impossible in certain cases [14]. A weaker question is whether one can construct global foliations such that the mean curvature of each leaf does not change sign.…”

Minimal area surfaces and fibered hyperbolic $3$-manifolds

“…Nevertheless, our results are "hands on" in nature and provide information about how curves of short (complex) length in a hyperbolic 3-manifold effect least area minimal immersions. Our main technical result provides a short argument for an improved version of a main result of [HW19]. This improvement allows us to easily construct the examples given in Theorems 1.1 and 1.2.…”

Minimal Surfaces in Hyperbolic 3‐Manifolds

“…In an earlier version of this paper, we used the tube lemma in our construction. The statement was basically that a least area surface cannot come very close to the core circle of a Margulis tube [19,22]. Thanks to the referee and Brian Bowditch, we realized a technical problem in our generalization, and we removed the tube lemma from our construction.…”