Cusp closing in rank one symmetric spaces

We investigate conditions under which cusps of pinched negative curvature can be closed as manifolds or orbifolds with nonpositive sectional curvature. We show that all cusps of complex hyperbolic type can be closed in this way whereas cusps of quaternionic or Cayley hyperbolic type cannot be closed. For cusps of real hyperbolic type we derive necessary and sufficient closing conditions. In this context we prove that a noncompact finite volume quotient of a rank one symmetric space can be approximated in the Gromov Hausdorff topology by closed orbifolds with nonpositive curvature if and only if it is real or complex hyperbolic. Using cusp closing methods we obtain new examples of real analytic manifolds of nonpositive sectional curvature and rank one containing flats. By the same methods we get an explicit resolution of the singularities in the Baily–Borel resp.Siu–Yau compactification of finite volume quotients of the complex hyperbolic space. Oblatum 2-IX-1994 & 7-VIII-1995     

Closed geodesics on orbifolds of nonpositive or nonnegative curvature

In this note we prove existence of closed geodesics of positive length on compact developable orbifolds of nonpositive or nonnegative curvature. We also include a geometric proof of existence of closed geodesics whenever the orbifold fundamental group contains a hyperbolic element and therefore reduce the existence problem to developable orbifolds with \(\pi _1^{orb}\) infinite and having finite exponent and finitely many conjugacy classes.     

Kodaira dimensions and hyperbolicity of nonpositively curved compact Kähler manifolds

In this article, we prove that a compact Kähler manifold M ⁿ with real analytic metric and with nonpositive sectional curvature must have its Kodaira dimension, its Ricci rank and the codimension of its Euclidean de Rham factor all equal to each other. In particular, M ⁿ is of general type if and only if it is without flat de Rham factor. By using a result of Lu and Yau, we also prove that for a compact Kähler surface M ² with nonpositive sectional curvature, if M ² is of general type, then it is Kobayashi hyperbolic.     

Projective deformations of hyperbolic Coxeter 3-orbifolds

By using Klein??s model for hyperbolic geometry, hyperbolic structures on orbifolds or manifolds provide examples of real projective structures. By Andreev??s theorem, many 3-dimensional reflection orbifolds admit a finite volume hyperbolic structure, and such a hyperbolic structure is unique. However, the induced real projective structure on some such 3-orbifolds deforms into a family of real projective structures that are not induced from hyperbolic structures. In this paper, we find new classes of compact and complete hyperbolic reflection 3-orbifolds with such deformations. We also explain numerical and exact results on projective deformations of some compact hyperbolic cubes and dodecahedra.     

Hyperbolic 3-manifolds with geodesic boundary: Enumeration and volume calculation

We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be applied in order to analyze simultaneously compact manifolds and finite-volume manifolds with toric cusps. In contrast, we show that if one allows annular cusps, the number of manifolds grows very rapidly and our strategy cannot be employed to obtain a complete list. We also carefully describe how to compute the volume of our manifolds, discussing formulas for the volume of a tetrahedron with generic dihedral angles in hyperbolic space.     

Very Ampleness of Line Bundles and Canonical Embedding of Coverings of Manifolds

Let L be an ample line bundle on a Kähler manifolds of nonpositive sectional curvature with K as the canonical line bundle. We give an estimate of m such that K+mL is very ample in terms of the injectivity radius. This implies that m can be chosen arbitrarily small once we go deep enough into a tower of covering of the manifold. The same argument gives an effective Kodaira Embedding Theorem for compact Kähler manifolds in terms of sectional curvature and the injectivity radius. In case of locally Hermitian symmetric space of noncompact type or if the sectional curvature is strictly negative, we prove that K itself is very ample on a large covering of the manifold.     

Pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature

In this paper, we use heat flow method to prove the existence of pseudo-harmonic maps from closed pseudo-Hermitian manifolds to Riemannian manifolds with nonpositive sectional curvature, which is a generalization of Eells–Sampson’s existence theorem. Furthermore, when the target manifold has negative sectional curvature, we analyze horizontal energy of geometric homotopy of two pseudo-harmonic maps and obtain that if the image of a pseudo-harmonic map is neither a point nor a closed geodesic, then it is the unique pseudo-harmonic map in the given homotopic class. This is a generalization of Hartman’s theorem.     

Properly discontinuous actions on Hilbert manifolds

In this article we study properly discontinuous actions on Hilbert manifolds giving new examples of complete Hilbert manifolds with nonnegative, respectively nonpositive, sectional curvature with infinite fundamental group. We also get examples of complete infinite dimensional Kähler manifolds with positive holomorphic sectional curvature and infinite fundamental group in contrastwith the finite dimensional case and we classify abelian groups acting linearly, isometrically and properly discontinuously on Stiefel manifolds. Finally, we classify homogeneous Hilbert manifolds with constant sectional curvature.     

Invariant subsets of rank 1 manifolds

It is proved that for a Riemannian manifold M with nonpositive sectional curvature and finite volume the space of directions at each point in which geodesic rays avoid a sufficiently small neighborhood of a fixed rank 1 vector v∈UM looks very much like a generalized Sierpinski carpet. We also show for nonpositively curved manifolds M with dim M≥ 3 the existence of proper closed flow invariant subsets of the unit tangent bundle UM whose footpoint projection is the whole of M. Received: 6 July 2000 / Revised version: 11 October 2001     

Partially hyperbolic geodesic flows

We construct a category of examples of partially hyperbolic geodesic flows which are not Anosov, deforming the metric of a compact locally symmetric space of nonconstant negative curvature. Candidates for such an example as the product metric and locally symmetric spaces of nonpositive curvature with rank bigger than one are not partially hyperbolic. We prove that if a metric of nonpositive curvature has a partially hyperbolic geodesic flow, then its rank is one. Other obstructions to partial hyperbolicity of a geodesic flow are also analyzed.     

Orbihedra of nonpositive curvature

A 2-dimensional orbihedron of nonpositive curvature is a pair (X, Γ), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and Γ is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X, Γ) which turns out to depend only on Γ and prove that, if X is boundaryless, then either (X, Γ) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. Partially supported by MSRI, SFB256 and University of Maryland. Partially supported by MSRI, SFB256 and NSF DMS-9104134.     

Examples of 4-manifolds with almost nonpositive curvature

We construct an infinite family of non-homeomorphic 4-manifolds with almost nonpositive sectional curvature whose universal covering space is not contractible. As a consequence, these manifolds do not support metrics with nonpositive sectional curvature. To achieve this, we use a generalization of Bavard's surgery construction, combined with an open book decomposition and knot theory.     

Structure of flat subspaces in low-dimensional manifolds of nonpositive curvature

The complete structure of flat subspaces is determined for compact real analytic 3 or 4-dimensional manifolds of nonpositive sectional curvature. In particular we show that this structure is determined by the fundamental group.     

Cusp closing of hyperbolic manifolds

A general method to produce infinitely many pairwise homotopically nonequivalent closed manifolds using the cusp closing construction is presented. An infinite sequence of closed homotopically nonequivalent real analytic Riemannian 5-manifolds with uniformly bounded volumes and uniformly bounded nonpositive sectional curvatures, which are allowed to vanish along codimension two submanifolds only, is constructed using this method.Both authors were supported in part by the Grant R24000 from the International Science Foundation.     

Orbifold fibrations of Eschenburg spaces

Most of the few known examples of compact Riemannian manifolds with positive sectional curvature are the total space of a Riemannian submersion. In this article we show that this is true for all known examples, if we enlarge the category to orbifold fibrations. For this purpose we study all almost free isometric circle actions on positively curved Eschenburg spaces, which give rise to principle orbifold bundle structures, and we examine in detail their geometric properties. In particular, we obtain a new family of 6-dimensional orbifolds with positive sectional curvature whose singular locus consists of just two points.      

Hyperbolic Structures on the Configuration Space of Six Points in the Projective Line

The oriented configuration space X⁺₆ of six points on the real projective line is a noncompact three-dimensional manifold which admits a unique complete hyperbolic structure of finite volume with ten cusps. On the other hand, it decomposes naturally into 120 cells each of which can be interpreted as the set of equiangular hexagons with unit area. Similar hyperbolic structures can be obtained by considering nonequiangular hexagons so that the standard hyperbolic structure on X⁺₆ is at the center of a five parameter family of hyperbolic structures of finite volume. This paper contributes to investigations of the properties of this family. In particular, we exhibit two real analytic maps from the set of prescribed angles of hexagons into R¹⁰ whose components are the traces of the monodromies at the ten cusps. We show that this map has maximal rank 5 at the center.     

On convex projective manifolds and cusps

This study of properly or strictly convex real projective manifolds introduces notions of parabolic, horosphere and cusp. Results include a Margulis lemma and in the strictly convex case a thick–thin decomposition. Finite volume cusps are shown to be projectively equivalent to cusps of hyperbolic manifolds. This is proved using a characterization of ellipsoids in projective space.     

Rationality of geometric signatures of complete 4-manifolds

Summary According to [CG4]|and [CFG], the complete manifolds with bounded sectional curvature and finite volume admit positive rank F-structures near infinity. In this paper, we show that, in dimension four, if the manifolds also have bounded covering geometry near infinity, then there exist F-structures with special topological properties. F-structures with these properties cannot be constructed solely by means of the general methods in [CG4]|and [CFG]. Using these special properties we prove a conjecture of Cheeger-Gromov on the rationality of the geometric signatures in the four dimensional case.Oblatum 15-XI-1993This work was partially supported by NSF Grant NSF DMS 9204095.     

Homogeneous manifolds with nonpositive curvature operator

Homgeneous manifolds of nonpositive sectional curvature can be identified with a certain class of solvable Lie groups. We determine, which of these groups also admit metrics with nonpositive curvature operator; this class is smaller, but still contains many examples.     

Almost Complex Manifolds and Hyperbolicity

One of the sufficient conditions for a complex manifold to be (complete) hyperbolic (in the sense that its intrinsic pseudo-distance is a (complete) distance) is that it admits a (complete) Hermitian metric with holomorphic sectional curvature bounded above by a negative constant. The concept of hyperbolicity can be readily extended to almost complex manifolds. We will show that the above result for hyperbolicity can be generalized to the almost complex case. As an application, we prove that every point of an almost complex manifold has a complete hyperbolic neighborhood. In real dimension 4, this fact was established by Debalme and Ivashkovich [2] by a completely different method.