Bilipschitz Embeddings of Metric Spaces into Space Forms

The paper describes some basic geometric tools to construct bilipschitz embeddings of metric spaces into (finite-dimensional) Euclidean or hyperbolic spaces. One of the main results implies the following: If X is a geodesic metric space with convex distance function and the property that geodesic segments can be extended to rays, then X admits a bilipschitz embedding into some Euclidean space iff X has the doubling property, and X admits a bilipschitz embedding into some hyperbolic space iff X is Gromov hyperbolic and doubling up to some scale. In either case the image of the embedding is shown to be a Lipschitz retract in the target space, provided X is complete.     

A damped hyerbolic equation with critical exponent

The aim of this paper is twofold. First, we initiate a detailed study of the so-called X^s _θ spaces attached to a partial differential operator. This include localization, duality, microlocal representation, subelliptic estimates, solvability and L^p (L^q ) estimates. Secondly, we obtain some theorems on the unique continuation of solutions to semilinear second order hyperbolic equations across strongly pseudo-convex surfaces. These results are proved using some new L^p → L^q Carleman estimates, derived using the X^s _θ spaces. Our theorems cover the subcritical case; in the critical case, the problem remains open. Similar results hold for higher order partial differential operators, provided that characteristic set satisfies a curvature conditions.     

Hyperbolic rank and subexponential corank of metric spaces

We introduce a new quasi-isometry invariant corank X of a metric space X called subexponential corank. A metric space X has subexponential corank k if roughly speaking there exists a continuous map , T is a topological space, such that for each the set g ^-1(t) has subexponential growth rate in X and the topological dimension dimT = k is minimal among all such maps. Our main result is the inequality for a large class of metric spaces X including all locally compact Hadamard spaces, where rank _h X is the maximal topological dimension of among all CAT(—1) spaces Y quasi-isometrically embedded into X (the notion introduced by M. Gromov in a slightly stronger form). This proves several properties of rank _h conjectured by Gromov, in particular, that any Riemannian symmetric space X of noncompact type possesses no quasi-isometric embedding of the standard hyperbolic space H ⁿ with . Submitted: February 2001, Revised: October 2001.     

Summability Properties for Multiplication Operators on Banach Function Spaces

Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized Köthe duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with ${x \in X}Consider a couple of Banach function spaces X and Y over the same measure space and the space X ^Y of multiplication operators from X into Y. In this paper we develop the setting for characterizing certain summability properties satisfied by the elements of X ^Y . At this end, using the “generalized K?the duality” for Banach function spaces, we introduce a new class of norms for spaces consisting of infinite sums of products of the type xy with x ? X{x \in X} and y ? Y{y \in Y} .     

Rigidity of certain admissible pairs of rational homogeneous spaces of Picard number 1 which are not of the subdiagram type

Recently, Mok and Zhang (2019) introduced the notion of admissible pairs (X₀, X) of rational homogeneous spaces of Picard number 1 and proved rigidity of admissible pairs (X₀, X) of the subdiagram type whenever X₀ is nonlinear. It remains unsolved whether rigidity holds when (X₀, X) is an admissible pair NOT of the subdiagram type of nonlinear irreducible Hermitian symmetric spaces such that (X₀, X) is nondegenerate for substructures. In this article we provide sufficient conditions for confirming rigidity of such an admissible pair. In a nutshell our solution consists of an enhancement of the method of propagation of sub-VMRT (varieties of minimal rational tangents) structures along chains of minimal rational curves as is already implemented in the proof of the Thickening Lemma of Mok and Zhang (2019). There it was proven that, for a sub-VMRT structure \(\overline{\omega} : \mathscr{C}(S) \rightarrow S\) on a uniruled projective manifold \((X,\,{\cal K})\) equipped with a minimal rational component and satisfying certain conditions so that in particular S is “uniruled” by open subsets of certain minimal rational curves on X, for a “good” minimal rational curve ? emanating from a general point x ∈ S, there exists an immersed neighborhood N_? of ? which is in some sense “uniruled” by minimal rational curves. By means of the Algebraicity Theorem of Mok and Zhang (2019), S can be completed to a projective subvariety Z ? X. By the author’s solution of the Recognition Problem for irreducible Hermitian symmetric spaces of rank ? 2 (2008) and under Condition (F), which symbolizes the fitting of sub-VMRTs into VMRTs, we further prove that Z is the image under a holomorphic immersion of X₀ into X which induces an isomorphism on second homology groups. By studying ?*-actions we prove that Z can be deformed via a one-parameter family of automorphisms to converge to X₀ ? X. Under the additional hypothesis that all holomorphic sections in Γ(X₀, T_x∣x₀) lift to global holomorphic vector fields on X, we prove that the admissible pair (X₀, X) is rigid. As examples we check that (X₀, X) is rigid when X is the Grassmannian G(n, n) of n-dimensional complex vector subspaces of W ? ?²ⁿ, n ? 3, and when X₀ ? X is the La grangian Grassmannian consisting of Lagrangian vector subspaces of (W, σ) where σ is an arbitrary symplectic form on W.

     

Decomposition Methods in Banach Spaces via Supplemented Subspaces Resembling Pełczyński’s Decomposition Methods

Suppose that X and Y are Banach spaces complemented in each other with supplemented subspaces A and B. In 1996, W. T. Gowers solved the Schroeder–Bernstein problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we obtain some suitable conditions involving the spaces A and B to yield that X is isomorphic to Y or to provide that at least X ^m is isomorphic to Yⁿ for some m, n ∈ IN^*. So we get some decomposition methods in Banach spaces via supplemented subspaces resembling Pełczyński’s decomposition methods. In order to do this, we introduce several notions of Schroeder–Bernstein Quadruples acting on the spaces X, Y, A and B. Thus, we characterize them by using some Banach spaces recently constructed. Received: October 4, 2005.     

Products of Hyperbolic Metric Spaces

Let (X _i d _i ), i=1,2, be proper geodesic hyperbolic metric spaces. We give a general construction for a 'hyperbolic product' X ₁× _h X ₂ which is itself a proper geodesic hyperbolic metric space and examine its boundary at infinity.     

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 the curvatures of bounded complete spacelike hypersurfaces in the Lorentz–Minkowski space

In this paper we characterize the spacelike hyperplanes in the Lorentz–Minkowski space L ^{n +1} as the only complete spacelike hypersurfaces with constant mean curvature which are bounded between two parallel spacelike hyperplanes. In the same way, we prove that the only complete spacelike hypersurfaces with constant mean curvature in L ^{n +1} which are bounded between two concentric hyperbolic spaces are the hyperbolic spaces. Finally, we obtain some a priori estimates for the higher order mean curvatures, the scalar curvature and the Ricci curvature of a complete spacelike hypersurface in L ^{n +1} which is bounded by a hyperbolic space. Our results will be an application of a maximum principle due to Omori and Yau, and of a generalization of it. Received: 5 July 1999     

A Hodge-Type Theorem for Manifolds with Fibered Cusp Metrics

A manifold with fibered cusp metrics X can be considered as a geometrical generalization of locally symmetric spaces of \mathbbQ{\mathbb{Q}}-rank one at infinity. We prove a Hodge-type theorem for this class of Riemannian manifolds, i.e. we find harmonic representatives of the de Rham cohomology H ^p (X). Similar to the situation of locally symmetric spaces, these representatives are computed by special values or residues of generalized eigenforms of the Hodge–Laplace operator on Ω ^p (X).     

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     

A. D. Alexandrov’s problem for non-positively curved spaces in the sense of Busemann

This paper is the last of a series devoted to the solution of Alexandrov’s problem for non-positively curved spaces. Here we study non-positively curved spaces in the sense of Busemann. We prove that isometries of a geodesically complete connected at infinity proper Busemann space X are characterized as follows: If a bijection f: X → X and its inverse f ⁻¹ preserve distance 1, then f is an isometry.     

On Interpolation Spaces with the Gelfand-Phillips Property

In this paper we study interpolation spaces generated by some interpolation functors. We show that under some conditions for Banach couples X and Y the spaces dual to the orbits of elements are Gelfand—Philips spaces. Consequently, the ideal of nuclear operators from X to Y contains a copy of l¹. We give also an interpolation theorem for limited operators.     

The metrical interpretation of superreflexivity in banach spaces

The main result is a metrical characterization of superreflexivity in Banach spaces. A Banach spaceX is not superreflexive if and only ifX contains hyperbolic trees as a metric space. The notion of non-linear cotype in discussed.     

Limit constructions over Riemann surfaces and their parameter spaces, and the commensurability group actions

To any compact hyperbolic Riemann surface X, we associate a new type of automorphism group — called its commensurability automorphism group, ComAut(X). The members of ComAut(X) arise from closed circuits, starting and ending at X, where the edges represent holomorphic covering maps amongst compact connected Riemann surfaces (and the vertices represent the covering surfaces). This group turns out to be the isotropy subgroup, at the point represented by X (in $ T_\infty $), for the action of the universal commensurability modular group on the universal direct limit of Teichmüller spaces, $ T_\infty $. Now, each point of $ T_\infty $ represents a complex structure on the universal hyperbolic solenoid. We notice that ComAut(X) acts by holomorphic automorphisms on that complex solenoid. Interestingly, this action turns out to be ergodic (with respect to the natural measure on the solenoid) if and only if the Fuchsian group uniformizing X is arithmetic. Furthermore, the action of the commensurability modular group, and of its isotropy subgroups, on some natural vector bundles over $ T_\infty $, are studied by us.     

On extensions of Pełczyński’s decomposition method in Banach spaces

Let X and Y be Banach spaces such that each of them is isomorphic to a complemented subspace of the other. In 1996, W. T. Gowers solved the Schroeder-Bernstein Problem for Banach spaces by showing that X is not necessarily isomorphic to Y. In this paper, we give suitable conditions on finite sums of X and Y to yield that X^m is isomorphic to Yⁿ for some In other words, we obtain some extensions of the well-known Pełczyński decomposition method in Banach spaces. In order to do this, we introduce the notion of Nearly Schroeder-Bernstein Quadruples for Banach spaces and pose a Conjecture to characterise them. Received: 5 January 2005     

On spaces in which countably compact sets are closed, and hereditary properties

A space X is called C-closed if every countably compact subset of X is closed in X. We study the properties of C-closed spaces. Among other results, it is shown that countably compact C-closed spaces have countable tightness and under Martin's Axiom or 2^ω0<2^ω1, C-closed is equivalent to sequential for compact Hausdorff spaces. Furthermore, every hereditarily quasi-k Hausdorff space is Fréchet-Urysohn, which generalizes a theorem of Arhangel'sk in [4]. Also every hereditarily q-space is hereditarily of pointwise countable type and contains an open dense first countable subspace.     

Banach function subspaces of L of a vector measure and related Orlicz spaces

Given a vector measure ν with values in a Banach space X, we consider the space L¹(ν) of real functions which are integrable with respect to ν. We prove that every order continuous Banach function space Y continuously contained in L¹(ν) is generated via a certain positive map related to ν and defined on X* x M, where X* is the dual space of X and M the space of measurable functions. This procedure provides a way of defining Orlicz spaces with respect to the vector measure ν.     

Cone characterization of Grothendieck spaces and Banach spaces containing <Emphasis Type="Italic">c</Emphasis><Subscript>0</Subscript>

In this article we study the embeddability of cones in a Banach space X. First we prove that c ₀ is embeddable in X if and only if its positive cone c₀⁺{c_0^+} is embeddable in X and we study some properties of Banach spaces containing c ₀ in the light of this result. So, unlike with the positive cone of ℓ ₁ which is embeddable in any non-reflexive space, c₀⁺{c_0^+} has the same behavior as the whole space c ₀. In the second part of this article we give a characterization of Grothendieck spaces X according to the geometry of cones of X*. By these results we give a partial positive answer to a problem of J.H. Qiu concerning the geometry of cones.     

The Leech lattice and complex hyperbolic reflections

We construct a natural sequence of finite-covolume reflection groups acting on the complex hyperbolic spaces ℂH ¹³, ℂH ⁹ and ℂH ⁵, and show that the 9-dimensional example coincides with the largest of the groups of Mostow [11]. Our reflection groups arise as automorphism groups of certain Lorentzian lattices over the Eisenstein integers, and we obtain our largest example by using the complex Leech lattice in a manner inspired by Conway [5]. We also construct finite-covolume reflection groups on the quaternionic hyperbolic spaces ?H ⁷, ?H ⁵ and ?H ³, again using the Leech lattice, and apply results of Borcherds [4] to obtain automorphic forms for our groups. Oblatum 25-III-1999 & 2-IX-1999?Published online: 21 February 2000