On complete Riemannian manifolds with collapsed ends

Shen, Zhong Min

doi:10.2140/pjm.1994.163.175

Cited by 5 publications

(7 citation statements)

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“…10. Infranilmanifolds are horosphere quotients Z. Shen constructed in [She94] a pinched negatively curved warped product metric on the product of an arbitrary infranilmanifold and (0, ∞) so that the metric is complete near the ∞-end, but is incomplete at the 0-end. Here we modify Shen's construction to produce a complete pinched negatively curved metric on the product of any infranilmanifold with R.…”

Section: Tubular Neighborhood Of An Orbitmentioning

confidence: 99%

Classification of negatively pinched manifolds with amenable fundamental groups

Belegradek

Kapovitch

2006

Acta Math.

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Section: Tubular Neighborhood Of An Orbitmentioning

confidence: 99%

Classification of negatively pinched manifolds with amenable fundamental groups

Belegradek

Kapovitch

2006

Acta Math.

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“…The vector space H p (2) (M) is isomorphic to the p-dimensional (reduced) L 2 -cohomology of M. For background on L 2 -cohomology, we refer to [8], [ From [14,Theorem 2], N I is diffeomorphic to an infranilmanifold. The proof of [14, Theorem 2] uses the collapsing results of Cheeger, Fukaya and Gromov, as given for example in [1].…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…In particular, it uses Fukaya's fibration theorem, along with the fact that U I is Gromov-Hausdorff close to a ray which passes through it. Strictly speaking, as in the proof of [14,Theorem 2], one may have to shrink U I a bit in order to apply the fibration theorem.…”

Section: Proof Of Theoremmentioning

confidence: 99%

On the spectrum of a finite-volume negatively-curved manifold

Lott¹

2001

ajm

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We show that a noncompact manifold with bounded sectional curvature, whose ends are sufficiently Gromov-Hausdorff close to rays, has a finite dimensional space of square-integrable harmonic forms. In the special case of a finite-volume manifold with pinched negative sectional curvature, we show that the essential spectrum of the p-form Laplacian is the union of the essential spectra of a collection of ordinary differential operators associated to the ends. We give examples of such manifolds with curvature pinched arbitrarily close to −1 and with an infinite number of gaps in the spectrum of the function Laplacian.

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“…And we say that a cusp metric g on Q × R is an eventually warped cusp metric if g = e −2t h + dt 2 , for t < c, for some c ∈ R and a metric h on Q. I. Belegradek and V. Kapovitch [3] (see also [2]) show, based on earlier work by Z.M. Shen [34], that if Q is almost flat then Q × R admits an eventually warped cusp metric.Addendum to Theorem A. Let g be an eventually warped cusp metric on Q × R. If the sectional curvatures of g lie in (a, b), with a < −1 < b, then we can take M in Theorem A with sectional curvatures also in (a, b).…”

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confidence: 99%

Riemannian hyperbolization

Ontaneda

2020

Publ.math.IHES

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Classical flat geometry has formed part of basic human knowledge since ancient times. It is characterized by the almost universally known condition that the sum of the internal angles of a triangle △ is equal to π. We write Σ(△) = π. Other fundamental geometries are defined by replacing the equality Σ(△) = π by inequalities; thus positively curved geometries and negatively curved geometries are determined by the inequalities Σ(△) > π and Σ(△) < π, respectively, where △ runs over all small non-degenerate triangles in a space. It is natural then to try to find spaces that admit such geometries, and this task has been a driving force in Riemannian Geometry for many decades. But surprisingly there are not too many examples of smooth closed manifolds that support either a positively curved or a negatively curved metric. For instance, besides spheres, in dimensions ≥ 17 (and = 24) the only positively curved simply connected known examples are complex and quaternionic projective spaces. In negative curvature the situation is arguably more striking because negative curvature has been studied extensively in many different areas in mathematics. Indeed, from the ergodicity of their geodesic flow in Dynamical Systems to their topological rigidity in Geometric Topology; from the existence of harmonic maps in Geometric Analysis to the well-studied and greatly generalized algebraic properties of their fundamental groups, negatively curved Riemannian manifolds are the main object in many important and well-known results in mathematics. Yet the fact remains that very few examples of closed negatively curved Riemannian manifolds are known. Besides the hyperbolic ones (R, C, H, O), the other known examples are the Mostow-Siu examples (complex dimension 2) which are local branched covers of complex hyperbolic space (1980, [23]), the Gromov-Thurston examples (1987, [18]) which are branched covers of real hyperbolic ones, the exotic Farrell-Jones examples (1989, [12]) which are homeomorphic but not diffeomorphic to real hyperbolic manifolds (and there are other examples of exotic type), and the three examples of Deraux (2005, [10]) which are of the Mostow-Siu type in complex dimension 3. Hence, excluding the Mostow-Siu and Deraux examples (in dimensions 4 and 6, respectively), all known examples of closed negatively curved Riemannian manifolds are homeomorphic to either a hyperbolic one or a branched cover of a hyperbolic one. * The author was partially supported by a NSF grant.objects are matched by the richness and complexity of the singularities obtained, and hyperbolized smooth manifolds are very far from being Riemannian. Interestingly one can relax and lose even more regularity and consider negative curvature from the algebraic point of view, that is consider Gromov's hyperbolic groups, and it can be argued [26] that "almost every group" is hyperbolic. So, negative curvature is in some weak sense generic, but Riemannian negative curvature seems very scarce. It is natural then to inquire about the difference between the class of ...

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On complete Riemannian manifolds with collapsed ends

Abstract: We show that if a complete open manifold with bounded curvature and sufficiently small ends, then each end is an infranilend. Conversely, an open manifold with finitely many infranilends admits a complete metric with bounded curvature and arbitrarily small ends.

Cited by 5 publications

References 1 publication

Classification of negatively pinched manifolds with amenable fundamental groups

Classification of negatively pinched manifolds with amenable fundamental groups

On the spectrum of a finite-volume negatively-curved manifold

Riemannian hyperbolization

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