Alexandrov Spaces Research Articles

A splitting theorem for Alexandrov spaces

We use a notion of differentiability for functions on Alexandrov spaces and prove a splitting theorem for Alexandrov spaces admitting affine functions with such differentiability.

Minimal Volume Alexandrov Spaces

Closed hyperbolic manifolds are proven to minimize volume over all Alexandrov spaces with curvature bounded below by -1 in the same bilipschitz class. As a corollary compact convex cores with totally geodesic boundary are proven to minimize volume over all hyperbolic manifolds in the same bilipschitz class. Also, closed hyperbolic manifolds minimize volume over all hyperbolic cone manifolds in the same bilipschitz class with cone angles not greater than 2pi. The proof uses techniques developed by Besson-Courtois-Gallot. In 3 dimensions, this result provides a partial solution to a conjecture in Kleinian groups concerning acylindrical manifolds.

Ideals of quasi-ordered sets

We define the $ \scr A $ -ideals of a poset – or equally of a quasi-ordered set – for various collections of subsets $ \scr A $ and corresponding $ \scr A $ -ideal continuity for functions. This leads us to a choice-free $ \scr A $ -ideal continuous imbedding of a poset into a $ \scr B $ -join complete poset with an appropriate universal mapping property. Topological applications include the imbedding of Scott spaces and Alexandrov spaces into up-complete Scott spaces.

Regularity of limits of noncollapsing sequences of manifolds

We prove that iterated spaces of directions of a limit of a noncollapsing sequence of manifolds with lower curvature bound are topologically spheres. As an application we show that for any finite dimensional Alexandrov space X n with $ n \ge 5 $ there exists an Alexandrov space Y homeomorphic to X which cannot be obtained as such a limit.

Existence of closed quasigeodesic in Alexandrov spaces of some types

We will prove the existence of closed quasigeodesics in compact Alexandrov spaces which can be approximated by Riemannian manifolds in the Lipschitz sense. By applying it, we will prove that every convex hypersurface in Euclidean spaces has a closed quasigeodesic.

Scalar curvature of definable Alexandrov spaces

To any compact set definable in an o-minimal structure, we associate a signed mea- sure, called scalar curvature measure. This generalizes the concept of scalar curvature on Rie- mannian manifolds. The main result states that, if the definable set is an Alexandrov space with curvature bounded below by k, then the scalar curvature measure is bounded below by kmðm � 1Þ volmð�Þ , where m is the dimension of the space and volmð�Þ the m-dimensional volume. This is a non-trivial generalization of a fact from dierential geometry. The proof combines techniques from o-minimal theory and from Alexandrov space theory. The back- ground of the definition of scalar curvature measure is given in the second part of the paper, where it is related to integral geometry and expressed by geometric quantities.

Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces

We prove the compactness of the imbedding of the Sobolev space $W^{1,2}_0(\Omega)$ into $L^2(\Omega)$ for any relatively compact open subset $\Omega$ of an Alexandrov space. As a corollary, the generator induced from the Dirichlet (energy) form has discrete spectrum. The generator can be approximated by the Laplacian induced from the DC-structure on the Alexandrov space. We also prove the existence of the locally Holder continuous heat kernel.

Convergence of Alexandrov spaces and spectrum of Laplacian

Denote by A(n) the family of isometry classes of compact n-dimensional Alexandrov spaces with curvature ≥-1, and λk(M) the kth eigenvalue of the Laplacian on M∈A(n). We prove the continuity of λk:A(n)→R with respect to the GromovHausdorff topology for each k,n∈N, and moreover that the spectral topology introduced by Kasue-Kumura [7], [8] coincides with the Gromov-Hausdorff topology on A(n).

On tangent cones of Alexandrov spaces with curvature bounded below

The tangent cones of an inner metric Alexandrov space with finite Hausdorff dimension and a lower curvature bound are always inner metric spaces with nonnegative curvature. In this paper we construct an infinite-dimensional inner metric Alexandrov space of nonnegative curvature which has in one point a tangent cone whose metric is not an inner metric.

Diameters of spherical Alexandrov spaces and curvature one orbifolds

Let G be a closed, non-transitive subgroup of O(n+1), where n ≥ 2, and let Q = S/G. We will show that for each n there is a lower bound for the diameter of Q. If G is finite then Q is an orbifold of constant curvature one and an explicit lower bound can be given. For Coxeter groups, the resulting lower bound is independent of dimension. Otherwise, Q is a spherical Alexandrov space and we will show existence of a lower bound. In the process, we will compute some examples of quotient spaces and their diameters.

Scott spaces and sober spaces

The categories of sober spaces and generalized (i.e., not necessarily T0) Scott spaces are adjoint. We define upper- and lower-sober spaces; lower-sober spaces are a reflective subcategory of TOP that is intermediate to that of sober spaces. We establish why the sobrifications of Alexandrov spaces and of certain Scott spaces described by Mislove (1981) are Scott.

Collapsing vs. Positive Pinching

Let M be a closed simply connected manifold and 0 < $ \delta \le 1 $ . Klingenberg and Sakai conjectured that there exists a constant $ {i_0} = {i_0}(M,\delta) > 0 $ such that the injectivity radius of any Riemannian metric g on M with $ \delta \le {K_g} \le 1 $ can be estimated from below by i 0. We study this question by collapsing and Alexandrov space techniques. In particular we establish a bounded version of the Klingenberg-Sakai conjecture: Given any metric d 0 on M, there exists a constant $ {i_0} = {i_0}(M,{d_0},\delta) > 0 $ , such that the injectivity radius of any $ \delta $ -pinched d 0-bounded Riemannian metric g on M (i.e., $ {\rm dist}_g \le d_0 $ and $ \delta \le K_g \le 1) $ can be estimated from below by i 0. We also establish a continuous version of the Klingenberg-Sakai conjecture, saying that a continuous family of metrics on M with positively uniformly pinched curvature cannot converge to a metric space of strictly lower dimension.

Metric minimizing surfaces

Consider a two-dimensional surface in an Alexandrov space of curvature bounded above by k k . Assume that this surface does not admit contracting deformations (a particular case of such surfaces is formed by area minimizing surfaces). Then this surface inherits the upper curvature bound, that is, this surface has also curvature bounded above by k k , with respect to the intrinsic metric induced from its ambient space.

A. D. Alexandrov's length manifolds with one-sided bounded curvature

The paper is devoted to discovering relations between inner metrics of manifolds with sectional curvature bounded from above (from below) by some constant and topological (simplicial, piecewise-linear, or smooth) structures on such manifolds. Examples of Alexandrov spaces with one-sided bounded curvature having exotic properties are given. Bibliography: 6 titles.

Amenable isometry groups of Hadamard spaces

One of the ever recurring themes in the theory of nonpositively curved spaces is the relation between geometry and topology. Since the universal covering space of a complete space of nonpositive curvature is contractible, this amounts to relations between the geometry of the space (or its universal covering space) and the algebraic structure of its fundamental group. A paradigm is the well-known result of Avez [Av] that the fundamental group of a compact Riemannian manifold M of nonpositive sectional curvature has exponential growth if and only if M is not flat. Gromov observed that the proof of Avez yields a somewhat stronger statement: the fundamental group of M is amenable if and only if M is flat, see [G1, p. 93]. Zimmer generalized this to the case where M is complete and of finite volume [Zi]. Further generalizations were obtained by Anderson [An] and Burger and Schroeder [BS]. The objective of this paper is the discussion of these results under the weaker assumption that the underlying spaces are Alexandrov spaces of nonpositive curvature. For the investigation of relations between the fundamental group and the geometry of a space of nonpositive curvature, it is convenient to study the isometric action of the fundamental group on the universal covering space. In many situations, it is not necessary to assume that a group action arises in this particular way and one studies isometric actions on complete simply connected spaces of nonpositive curvature. This is the case in most of our arguments below. Let X be a Hadamard space, that is, a complete simply connected geodesic space of nonpositive curvature in the sense of Alexandrov. A flat in X is a

Parallel Transportation for Alexandrov Space with Curvature Bounded Below

In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and prove an analog of the second variation formula for this case. A closely related construction has been made for Alexandrov space with bilaterally bounded curvature by Igor Nikolaev (see [N]).¶Naturally, as we have a more general situation, the constructed transportation does not have such good properties as in the case of bilaterally bounded curvature. In particular, we cannot prove the uniqueness in any good sense. Nevertheless the constructed transportation is enough for the most important applications such as Synge's lemma and Frankel's theorem. Recently by using this parallel transportation together with techniques of harmonic functions on Alexandrov space, we have proved an isoperimetric inequality of Gromov's type.¶Author is indebted to Stephanie Alexander, Yuri Burago and Grisha Perelman for their willingness to understand, interest and important remarks.

Alexandrov spaces with nonnegative curvature outside a compact set

Alexandrov spaces with nonnegative curvature outside a compact set

Topological regularity theorems for Alexandrov spaces

Since Gromov gave in [G1], [G2] an abstract definition of Hausdorff distance between two compact metric spaces, the Gromov-Hausdorff convergence theory has played an important role in Riemannian geometry. Usually, Gromov-Haus- dorff limits of Riemannian manifolds are almost never Riemannian manifolds. This motivates the study of Alexandrov spaces which are more singular than Riemannian manifolds since it is observed in [GP1] that the limit spaces are Alexandrov spaces if the manifolds in the sequence have curvature bounded uni- formly from below. Alexandrov spaces are finite dimensional inner metric spaces with a lower curvature bound in the sense of distance comparison. It is now well known that the topological and geometric properties of Gromov-Hausdorff limits will reveal those of Riemannian manifolds considered in the sequence. For a discussion of this viewpoint, see [W1]. In view of this, the investiga- tion of the topological and geometric properties of Alexandrov spaces has re- cently attracted a lot of attention; see for example [BGP], [FY], [GP1], [Pe], [Sh] and [Pt]. The structure of Alexandrov spaces is studied in [BGP], [Pe] and [Pt]. In particuar, if P is a point in an Alexandrov space X , then the space of directions Σ p at P is an Alexandrov space of one less dimension and with curvature Zl. Moreover, a neighborhood of P in X is homeomorphic to the linear cone over Σ p . One important implication of this local structure result is that if Σ p is a sphere then the point P is a manifold point. However, the converse is not true. This can be seen from the following example from

Duality for semilattice representations

The paper presents general machinery for extending a duality between complete, cocomplete categories to a duality between corresponding categories of semilattice representations (i.e. sheaves over Alexandrov spaces). This enables known dualities to be regularized. Among the applications, regularized Lindenbaum-Tarski duality shows that the weak extension of Boolean logic (i.e. the semantics of PASCAL-like programming languages) is the logic for semilattice-indexed systems of sets. Another application enlarges Pontryagin duality by regularizing it to obtain duality for commutative inverse Clifford monoids.

Morse theory for Min-type functions

0. Introduction. Morse theory for distance functions was initiated by GroveShiohama [GS] and Gromov’s [G1] paper where basic notions of the theory were formulated; even this very initial level of the theory leads to important geometric applications. Grove and Shiohama [GS] established a generalized sphere theorem by constructing a vector field on a Riemannian manifold, with the property that the distance function had no stationary points along the integral curves at non-singular points of the vector field. They showed that this vector field had exactly two singular points and hence the manifold was homeomorphic to a sphere. Later Gromov [G1] was able to bound the sum of the Betti numbers of a positively curved Riemannian manifold. He controlled the location of critical points of a Riemannian distance function on a positively curved manifold using Toponogov’s theorem and then was able to bound the number of critical points of this function and hence the homology of the manifold, using a spectral sequence argument. Morse theory for Riemannian distance functions was discussed in [Gr], [L] and other papers, which explained its importance for geometric applications rather than developed the theory itself. Even a suitable concept of the index of a critical point has not been developed. As a result, the most powerful tools of the classical Morse theory such as Morse inequalities and the correspondence between critical points and the handle decomposition of the manifold cannot be used. The relationship with the classical Morse theory has not been investigated either; in particular, the connection between notions of the critical points in both theories is not obvious. In a different direction, the structure of Alexandrov spaces with curvatures bounded below was investigated using distance functions in several papers starting with [BGP] and continuing with [P1], [P2]. A key result obtained is a canonical stratification of such Alexandrov spaces into topological manifolds and again the technique is a type of Morse theory, using the distance function. As Alexandrov spaces do not have as much structure as Riemannian manifolds, our theory gives more detailed information on the nature of critical points and index. M. Gromov pointed out in [G1] that the Morse theory for Riemannian distance functions can be developed by analogy with the classical Morse theory. The aim of this paper is to construct a Morse theory for functions which are minima of finite families of smooth functions and clarify the connection with Riemannian distance functions for non-positively curved manifolds. We develop the theory in the classical style, including the notion of the index for critical points, and clarify relations with the Grove-Shiohama-Gromov approach.