Surfaces In Hyperbolic 3-manifolds Research Articles

A nonexistence result for CMC surfaces in hyperbolic 3-manifolds

We prove that a complete hyperbolic 3-manifold of finite volume does not admit a properly embedded noncompact surface of finite topology with constant mean curvature greater than or equal to 1.

Curvature Bounds for Surfaces in Hyperbolic 3-Manifolds

AbstractA triangulation of a hyperbolic 3-manifold is L-thick if each tetrahedron having all vertices in the thick part of M is L-bilipschitz diffeomorphic to the standard Euclidean tetrahedron. We show that there exists a fixed constant L such that every complete hyperbolic 3-manifold has an L-thick geodesic triangulation. We use this to prove the existence of universal bounds on the principal curvatures of π1-injective surfaces and strongly irreducible Heegaard surfaces in hyperbolic 3-manifolds.

Renormalized Area and Properly Embedded Minimal Surfaces in Hyperbolic 3-Manifolds

We study the renormalized area functional \({\mathcal{A}}\) in the AdS/CFT correspondence, specifically for properly embedded minimal surfaces in convex cocompact hyperbolic 3-manifolds (and somewhat more broadly, Poincare-Einstein spaces). Our main results include an explicit formula for the renormalized area of such a minimal surface Y as an integral of local geometric quantities, as well as formulae for the first and second variations of \({\mathcal{A}}\) which are given by integrals of global quantities over the asymptotic boundary loop γ of Y. All of these formulae are also obtained for a broader class of nonminimal surfaces. The proper setting for the study of this functional (when the ambient space is hyperbolic) requires an understanding of the moduli space of all properly embedded minimal surfaces with smoothly embedded asymptotic boundary. We show that this moduli space is a smooth Banach manifold and develop a \({\mathbb{Z}}\) -valued degree theory for the natural map taking a minimal surface to its boundary curve. We characterize the nondegenerate critical points of \({\mathcal{A}}\) for minimal surfaces in \({\mathbb{H}^3}\) , and finally, discuss the relationship of \({\mathcal{A}}\) to the Willmore functional.

Thick Surfaces in Hyperbolic 3-Manifolds

We show that every closed, virtually fibered hyperbolic 3-manifold contains immersed, quasi-Fuchsian.

Shrinkwrapping and the taming of hyperbolic 3-manifolds

We introduce a new technique for finding CAT(-1) surfaces in hyperbolic 3-manifolds. We use this to show that a complete hyperbolic 3-manifold with finitely generated fundamental group is geometrically and topologically tame.

Cusp size bounds from singular surfaces in hyperbolic 3-manifolds

Singular maps of surfaces into a hyperbolic 3-manifold are utilized to find upper bounds on meridian length, ℓ \ell -curve length and maximal cusp volume for the manifold. This allows a proof of the fact that there exist hyperbolic knots with arbitrarily small cusp density and that every closed orientable 3-manifold contains a knot whose complement is hyperbolic with maximal cusp volume less than or equal to 9. We also find particular upper bounds on meridian length, ℓ \ell -curve length and maximal cusp volume for hyperbolic knots in S 3 \mathbb {S}^3 depending on crossing number. Particular improved bounds are obtained for alternating knots.

Immersed essential surfaces in hyperbolic 3-manifolds

Immersed essential surfaces in hyperbolic 3-manifolds

On a theorem of Scott and Swarup

Let 1 → H → G → Z → 1 1 \rightarrow H \rightarrow G \rightarrow \mathbb {Z} \rightarrow 1 be an exact sequence of hyperbolic groups induced by an automorphism ϕ \phi of the free group H H . Let H 1 ( ⊂ H ) H_1 ( \subset H) be a finitely generated distorted subgroup of G G . Then there exist N > 0 N > 0 and a free factor K K of H H such that the conjugacy class of K K is preserved by ϕ N \phi ^N and H 1 H_1 contains a finite index subgroup of a conjugate of K K . This is an analog of a theorem of Scott and Swarup for surfaces in hyperbolic 3-manifolds.

Immersed geodesic surfaces in hyperbolic 3-manifolds

A closed geodesic in a hyperbolic surface with either a short common orthogonal or many self-intersections is uniformly long. We will prove analogous results for immersed geodesic surfaces in hyperbolic 3-manifolds, that the length of common orthogonal admits a universal lower bound, and that the size of self-intersection singularities has a universal upper bound, both depending only on the area of surfaces.

Totally geodesic surfaces in hyperbolic 3-manifolds

In this paper we investigate totally geodesic surfaces in hyperbolic 3-manifolds. In particular we show that if M is a compact arithmetic hyperbolic 3-manifold containing an immersion of a totally geodesic surface then it contains infinitely many commensurability classes of such surfaces. In addition we show for these M that the Chern-Simons invariant is rational.We also show, that unlike the figure-eight knot complement in S3, many knot complements in S3 do not contain an immersion of a closed totally geodesic surface.

Incompressible surfaces in punctured-torus bundles

We derive in this paper the classification up to isotopy of the incompressible surfaces in hyperbolic 3-manifolds which fiber over the circle with fiber a once-punctured torus. From this classification it follows that most of the 3-manifolds obtained by compactifying these bundles via a circle at infinity are closed hyperbolic 3-manifolds which contain 1.0 incompressible surfaces, i.e., are not Haken manifolds.