Cusped Hyperbolic 3-manifolds Research Articles

Closed minimal surfaces in cusped hyperbolic three-manifolds

Motivated by classical theorems on minimal surface theory in compact hyperbolic 3-manifolds, we investigate the questions of existence and deformations for least area minimal surfaces in complete noncompact hyperbolic 3-manifold of finite volume. We prove any closed immersed incompressible surface can be deformed to a closed immersed least area surface within its homotopy class in any cusped hyperbolic 3-manifold. Our techniques highlight how special structures of these cusped hyperbolic 3-manifolds prevent any least area minimal surface going too deep into the cusped region.

1–efficient triangulations and the index of a cusped hyperbolic 3–manifold

In this paper we will promote the 3D index of an ideal triangulation T of an oriented cusped 3-manifold M (a collection of q-series with integer coefficients, introduced by Dimofte-Gaiotto-Gukov) to a topological invariant of oriented cusped hyperbolic 3-manifolds. To achieve our goal we show that (a) T admits an index structure if and only if T is 1-efficient and (b) if M is hyperbolic, it has a canonical set of 1-efficient ideal triangulations related by 2-3 and 0-2 moves which preserve the 3D index. We illustrate our results with several examples.

Cusped hyperbolic 3-manifolds of complexity 10 having maximum volume

We give a complete census of orientable cusped hyperbolic 3-manifolds obtained by gluing at most ten regular ideal hyperbolic tetrahedra. Although the census is exhaustive, the question of nonhomeomorphism remains open for some pairs of manifolds with one, two, and three cusps.

Dehn Filling of Cusped Hyperbolic 3-Manifolds with Geodesic Boundary

We define for each $g \geq 2$ and $k \geq 0$ a set $\mathcal{M}_{g,k}$ of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus $g$. Manifolds in $\mathcal{M}_{g,k}$ have Matveev complexity $g + k$ and Heegaard genus $g+1$, and their homology, volume, and Turaev-Viro invariants depend only on $g$ and $k$. In addition, they do not contain closed essential surfaces. The cardinality of $\mathcal{M}_{g,k}$ for a fixed $k$ has growth type $g^g$. We completely describe the non-hyperbolic Dehn fillings of each $M$ in $\mathcal{M}_{g,k}$, showing that, on any cusp of any hyperbolic manifold obtained by partially filling $M$, there are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with $\partial$-reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If $M$ has one cusp only, the three $\partial$-reducible fillings are handlebodies.

Chern-Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space ofT 2

We describe a cut-and-paste method for computing Chern-Simons invariant of flatG-connections on 3-manifolds decomposed along tori, especially forG=SU(2) andSL(2,C). We use this method to make computations ofSU(2) Chern-Simons invariants of graph manifolds which generalize Fintushel and Stern's computations for Seifert-fibered spaces. We also use this technique to give a simple derivation of a formula of Yoshida relating the flatSL(2,C) Chern-Simons invariant of the holonomy representation to the volume and the metric Chern-Simons invariant for cusped hyperbolic 3-manifolds.