Detection of incompressible surfaces in hyperbolic punctured torus bundles

Culler and Shalen, and later Yoshida, give ways to construct incompressible surfaces in 3-manifolds from ideal points of the character and deformation varieties, respectively. We work in the case of hyperbolic punctured torus bundles, for which the incompressible surfaces were classified by Floyd and Hatcher, and independently by Culler, Jaco and Rubinstein. We convert non fiber incompressible surfaces from their form to the form output by Yoshida’s construction, and run his construction backwards to give (for non semi-fibers, which we identify) the data needed to construct ideal points of the deformation variety corresponding to those surfaces via Yoshida’s construction. We use a result of Tillmann to show that the same incompressible surfaces can be obtained from an ideal point of the character variety via the Culler-Shalen construction. In particular this shows that all boundary slopes of non fiber and non semi-fiber incompressible surfaces in hyperbolic punctured torus bundles are strongly detected.     

Hyperbolic surfaces in the Grassmannian

In this article we study real 2-dimensional surfaces in the Grassmannian of 2-planes in a 4-dimensional vector space. These surfaces occur naturally as the fibers of jet bundles of partial differential equations.On the Grassmannian there is an invariant conformal quadratic form and we will use the structure induced by this quadratic form to study the surfaces. We give a topological classification of compact hyperbolic surfaces similar to the classification by Gluck and Warner [Duke Math. J. 50 (1) (1983)] of compact elliptic surfaces. In contrast with elliptic surfaces there are several topological possibilities for hyperbolic surfaces. We make a calculation of the differential invariants under the action of the group of conformal isometries. Finally, we analyze a class of surfaces called geometrically flat and show that within this class there exist many examples of non-trivial compact surfaces.     

Some curious Kleinian groups and hyperbolic 5-manifolds

We present a new family of discrete subgroups ofSO (5, 1) isomorphic to lattices inSO (3, 1). In some of the examples the limit sets are wildly knotted 2-spheres. As an application we produce complete hyperbolic 5-manifolds that are nontrivial plane bundles over closed hyperbolic 3-manifolds and conformally flat 4-manifolds that are nontrivial circle bundles over closed hyperbolic 3-manifolds.     

On a class of hyperbolic 3-manifolds and groups with one defining relation

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, arbitrarily many of the same volume. The fundamental groups of these 3-manifolds are groups with one defining relation. Our main result is a classification of these manifolds up to homeomorphism, resp. isometry.     

3-manifolds whose universal coverings are Lie groups

We classify those closed 3-manifolds whose universal covering space naturally admits the structure of a Lie group     

The curve complex and covers via hyperbolic 3-manifolds

Rafi and Schleimer recently proved that the natural relation between curve complexes induced by a covering map between two surfaces is a quasi-isometric embedding. We offer another proof of this result using a distance estimate via hyperbolic 3-manifolds.     

Complete stable CMC surfaces with empty singular set in Sasakian sub-Riemannian 3-manifolds

For constant mean curvature surfaces of class C ² immersed inside Sasakian sub-Riemannian 3-manifolds we obtain a formula for the second derivative of the area which involves horizontal analytical terms, the Webster scalar curvature of the ambient manifold, and the extrinsic shape of the surface. Then we prove classification results for complete surfaces with empty singular set which are stable, i.e., second order minima of the area under a volume constraint, inside the 3-dimensional sub-Riemannian space forms. In the first Heisenberg group we show that such a surface is a vertical plane. In the sub-Riemannian hyperbolic 3-space we give an upper bound for the mean curvature of such surfaces, and we characterize the horocylinders as the unique ones with squared mean curvature 1. Finally we deduce that any complete surface with empty singular set in the sub-Riemannian 3-sphere is unstable.     

Generalized Markoff maps and McShane's identity

We study the (relative) SL(2,C) character varieties of the one-holed torus and the action of the mapping class group on the (relative) character variety. We show that the subset of characters satisfying two simple conditions called the Bowditch Q-conditions is open in the relative character variety and that the mapping class group acts properly discontinuously on this subset. Furthermore, this is the largest open subset for which this holds. We also show that a generalization of McShane's identity holds for all characters satisfying the Bowditch Q-conditions. Finally, we show that further variations of the McShane-Bowditch identity hold for characters which are fixed by an Anosov element of the mapping class group and which satisfy a relative version of the Bowditch Q-conditions, with applications to identities for incomplete hyperbolic structures on punctured torus bundles over the circle, and also for closed hyperbolic 3-manifolds which are obtained by hyperbolic Dehn surgery on such manifolds.     

A note on the Chern-Simons invariant of hyperbolic 3-manifolds

In this note we study how the Chern-Simons invariant behaves when two hyperbolic 3-manifolds are glued together along incompressible
thrice-punctured spheres. Such an operation produces many hyperbolic 3-manifolds with different numbers of cusps sharing the same volume and the same Chern-Simons invariant. The results in this note, combined with those of Meyerhoff and Ruberman, give an algorithm for determining the unknown constant in Neumann's simplicial formula for the Chern-Simons invariant of hyperbolic 3-manifolds.

     

All flat three-manifolds appear as cusps of hyperbolic four-manifolds

There are ten diffeomorphism classes of compact, flat 3-manifolds. It has been conjectured that each of these occurs as the boundary of a 4-manifold whose interior admits a complete, hyperbolic structure of finite volume. This paper provides evidence in support of the conjecture. In particular, each diffeomorphism class of compact, flat 3-manifolds is shown to appear as one of the cusps of a complete, finite-volume, hyperbolic 4-manifold. This is done with a construction that uses special coverings of ³ by 3-balls. A further consequence of the construction is a finer result about the geometric structures which can be induced on cusps of complete, finite-volume, hyperbolic 4-manifolds. Using Mostow's Rigidity Theorem, one can show that not every flat structure occurs in this way. However, the fact that the flat structures induced on cusps of such 4-manifolds are dense in their respective moduli spaces follows from the construction.     

Construction of pseudo-isometries for treelike hyperbolic 3-manifolds of infinite volume

We introduce a family of rigid hyperbolic 3-manifolds of infinite volume with possibly infinitely many ends: the treelike manifolds. These manifolds generalize a family of constructive non compact surfaces – the equational surfaces – for which the homeomorphism problem is decidable. The proof of rigidity relies firstly on Thurston's theorem of compactness of the Teichmüller space of acylindrical compact 3-manifolds, and secondly, on Sullivan's rigidity theorem. To cite this article: O. Ly, C. R. Acad. Sci. Paris, Ser. I 337 (2003).     

A note on hyperbolic 3-manifolds of the same volume

We describe a method for constructing an arbitrary number of closed hyperbolic 3-manifolds of the same volume. In fact we prove that many hyperbolic 3-manifolds of finite volume have an arbitrary number of non-homeomorphic finite convering spaces of the same degree and hence the same volume. This applies, for example, to all hyperbolic 3-manifolds whose universal covering group is a subgroup of finite index in a Coxeter group generated by the reflections in the faces of a hyperbolic Coxeter polyhedron. It also applies to all hyperbolic 3-manifolds of finite volume with at least one cusp.     

A remark on renormalized volume and Euler characteristic for ache 4-manifolds

This note computes a renormalized volume and a renormalized Gauss-Bonnet-Chern formula for asymptotically complex hyperbolic Einstein (so-called ache) 4-manifolds.     

Hyperbolic 3-manifolds as cyclic branched coverings

There is an extensive literature on the characterization of knots in the 3-sphere which have the same 3-manifold as a common n-fold cyclic branched covering, for some integer . In the present paper, we study the following more general situation. Given two integers m and n, how are knots K ₁ and K ₂ related such that the m-fold cyclic branched covering of K ₁ coincides with the n-fold cyclic branched covering of K ₂. Or, seen from the point of view of 3-manifolds: in how many different ways can a given 3-manifold occur as a cyclic branched covering of knots in S ³. Under certain hypotheses, we solve this problem for the basic class of hyperbolic 3-manifolds and hyperbolic knots (the other basic class is that of Seifert fiber spaces resp. of torus and Montesinos knots for which the situation is well understood; the general case can then be analyzed using the equivariant sphere and torus decomposition into Seifert fiber spaces and hyperbolic manifolds). Received: December 7, 1999; revised version: May 22, 2000     

Small dilatation pseudo-Anosov homeomorphisms and 3-manifolds

The main result of this paper is a universal finiteness theorem for the set of all small dilatation pseudo-Anosov homeomorphisms ?:S→S, ranging over all surfaces S. More precisely, we consider pseudo-Anosov homeomorphisms ?:S→S with |χ(S)|log(λ(?)) bounded above by some constant, and we prove that, after puncturing the surfaces at the singular points of the stable foliations, the resulting set of mapping tori is finite. Said differently, there is a finite set of fibered hyperbolic 3-manifolds so that all small dilatation pseudo-Anosov homeomorphisms occur as the monodromy of a Dehn filling on one of the 3-manifolds in the finite list, where the filling is on the boundary slope of a fiber.     

On the weak discontinuity surfaces and tapes of equation systems in fluid mechanics

The weak discontinuity surfaces for a system of quasi-linear differential equations of higher order are developed. The classification of equation systems in fluid mechanics is based on the propagative weak discontinuity surfaces. Types of equations for different flow models are discussed. The conclusion is as follows:(a) For incompressible nonviscous flow, incompressible viscous flow and compressible viscous flow, the types of equations are all parabolic in the unsteady case and elliptic in the steady case.(b) For compressible nonviscous flow, the type of equations is hyperbolic in the unsteady case or steady supersonic case, and the type is elliptic in the steady subsonic case.     

闭曲面丛的一类曲面连通和中的本质闭曲面

In this paper, we will characterize all types of essential closed surfaces in a class of surface sum ofI-bundle of closed surfaces, and give an application of the classificatioa in the surface sum of two 3-manifolds.     

Bounded gaps between primes and the length spectra of arithmetic hyperbolic 3-orbifolds

In 1992, Reid asked whether hyperbolic 3-manifolds with the same geodesic length spectra are necessarily commensurable. While this is known to be true for arithmetic hyperbolic 3-manifolds, the non-arithmetic case is still open. Building towards a negative answer to this question, Futer and Millichap recently constructed infinitely many pairs of non-commensurable, non-arithmetic hyperbolic 3-manifolds which have the same volume and whose length spectra begin with the same first m geodesic lengths. In the present paper, we show that this phenomenon is surprisingly common in the arithmetic setting. In particular, given any arithmetic hyperbolic 3-orbifold derived from a quaternion algebra, any finite subset S of its geodesic length spectrum, and any $k\ge 2$, we produce infinitely many k-tuples of arithmetic hyperbolic 3-orbifolds which are pairwise non-commensurable, have geodesic length spectra containing S, and have volumes lying in an interval of (universally) bounded length. The main technical ingredient in our proof is a bounded gaps result for prime ideals in number fields lying in Chebotarev sets which extends recent work of Thorner.     

Singhof fillings of closed 3-manifolds

We give a complete classification of all closed, connected 3-manifolds which admit a Singhof filling with any number of solid tori. Received: 15 March 2001 / Revised version: 17 September 2001     

Reducing Dehn Fillings and Small Surfaces

In this paper we investigate the distances between Dehn fillingson a hyperbolic 3-manifold that yield 3-manifolds containingessential small surfaces including non-orientable surfaces.In particular, we study the situations where one filling createsan essential sphere or projective plane, and the other createsan essential sphere, projective plane, annulus, Möbiusband, torus or Klein bottle, for all eleven pairs of such non-hyperbolicmanifolds. 2000 Mathematics Subject Classification 57M50.