Simplices of Maximal Volume or Minimal Total Edge Length in Hyperbolic Space

In this paper, we are mainly concerned with n-dimensional simplicesin hyperbolic space Hⁿ. We will also consider simplices withideal vertices, and we suggest that the reader keeps the Poincaréunit ball model of hyperbolic space in mind, in which the sphereat infinity Hⁿ() corresponds to the bounding sphere of radius1. It is known that all hyperbolic simplices (even the idealones) have finite volume. However, explicit calculation of theirvolume is generally a very difficult problem (see, for example,[1] or [16]). Our first theorem states that, amongst all simplicesin a closed geodesic ball, the simplex of maximal volume isregular. We call a simplex regular if every permutation of itsvertices can be realized by an isometry of Hⁿ. A correspondingresult for simplices in the sphere has been proved by Böröczky[4].     

Type 2 Semi-Algebras of Continuous Functions

A semi-algebra of continuous functions is a cone A of continuousreal functions on a compact Hausdorff space X such that A containsthe products of its elements. A cone A is said to be of typen if fA implies fⁿ(1 + f)^–1 A. Uniformly closed semi-algebrasof types 0 and 1 have long been characterized in a manner analogousto the Stone–Weierstrass theorem, but, except for thecase when A is generated by a single function, little has beenknown about type 2. Here, progress is reported on two problems.The first is the characterization of those continuous linearfunctionals on C(X) that determine semi-algebras of type 2.The second is the determination of the type of the tensor productof two type 1 semi-algebras. 1991 Mathematics Subject Classification:46J10.     

On splitting theorems for CAT(0) spaces and compact geodesic spaces of non-positive curvature

In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group Γ is said to be rigid, if Γ determines its boundary up to homeomorphisms of a CAT(0) space on which Γ acts geometrically. C. Croke and B. Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if Γ₁ and Γ₂ are rigid CAT(0) groups then so is Γ₁ × Γ₂.     

The eigenvalues of limits of radial Toeplitz operators

Let A² be the Bergman space on the unit disk. A bounded operatorS on A² is called radial if Szⁿ = _n zⁿ for all n 0, where _nis a bounded sequence of complex numbers. We characterize theeigenvalues of radial operators that belong to the Toeplitzalgebra.     

Damping Oscillatory Integrals with Polynomial Phase and Convolution Operators with the Affine Arclength Measure on Polynomial Curves in Rn

McMichael proved that the convolution with the (euclidean) arclengthmeasure supported on the curve t (t, t², ..., tⁿ), 0 < t< 1, maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if and only if 2n(n+1)/(n²+n+2) p 2. In proving this, a uniform estimate on damping oscillatoryintegrals with polynomial phase was crucial. In this paper,a remarkably simple proof of the same estimate on oscillatoryintegrals is presented. In addition, it is shown that the convolutionoperator with the affine arclength measure on any polynomialcurve in Rⁿ maps L^p(Rⁿ) boundedly into L^p'(Rⁿ) if p = 2n(n+1)/(n²+n+2).     

Weyl's Theorem, a-Weyl's Theorem, and Local Spectral Theory

Necessary and sufficient conditions are given for a Banach spaceoperator with the single-valued extension property to satisfyWeyl's theorem and a-Weyl's theorem. It is shown that if T orT^* has the single-valued extension property and T is transaloid,then Weyl's theorem holds for f(T)for every fH((T)). When T^*has the single-valued extension property, T is transaloid andT is a-isoloid, then a-Weyl's theorem holds for f(T) for everyfH((T)). It is also proved that if T or T^* has the single-valuedextension property, then the spectral mapping theorem holdsfor the Weyl spectrum and for the essential approximate pointspectrum.     

Interpolation by Polynomials in z and z-1 on an Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space of functionsanalytic in the annulus r|z|R and continuous on its boundary.The points of interpolation are chosen to coincide with then roots of zⁿ=Rⁿeⁱⁿ (0<<2/n) and the n roots of zⁿ=rⁿ.The behaviour of the corresponding Lebesgue function is studied,and an estimate for the operator norm is obtained. The resultsof the present paper give a partial affirmative answer to twoconjectures suggested earlier by Mason on the basis of numericalcomputations.     

Local Rigidity of Infinite-Dimensional Teichmuller Spaces

This paper presents a rigidity theorem for infinite-dimensionalBergman spaces of hyperbolic Riemann surfaces, which statesthat the Bergman space A¹(M), for such a Riemann surface M,is isomorphic to the Banach space of summable sequence, l¹.This implies that whenever M and N are Riemann surfaces thatare not analytically finite, and in particular are not necessarilyhomeomorphic, then A¹(M) is isomorphic to A¹(N). It is knownfrom V. Markovic that if there is a linear isometry betweenA¹(M) and A¹(N), for two Riemann surfaces M and N of non-exceptionaltype, then this isometry is induced by a conformal mapping betweenM and N. As a corollary to this rigidity theorem presented here,taking the Banach duals of A¹(M) and l¹ shows that the spaceof holomorphic quadratic differentials on M, Q(M), is isomorphicto the Banach space of bounded sequences, l. As a consequenceof this theorem and the Bers embedding, the Teichmüllerspaces of such Riemann surfaces are locally bi-Lipschitz equivalent.     

The Bestvina-Brady Construction Revisited: Geometric Computation of {sum}-Invariants for Right-Angled Artin Groups

The starting point of our investigation is the remarkable paper[2] in which Bestvina and Brady gave an example of an infinitelyrelated group of type FP₂. The result about right-angled Artingroups behind their example is best interpreted by means ofthe Bieri–Strebel–Neumann–Renz -invariants. For a group G the invariants ⁿ(G) and ⁿ(G, Z) are sets of non-trivialhomomorphisms :GR. They contain full information about finitenessproperties of subgroups of G with abelian factor groups. Themain result of [2] determines for the canonical homomorphism, taking each generator of the right-angled Artin group G to1, the maximal n with ⁿ(G), respectively ⁿ(G, Z). In [6] Meier, Meinert and VanWyk completed the picture by computingthe full -invariants of right-angled Artin groups using as wellthe result of Bestvina and Brady as algebraic techniques from-theory. Here we offer a new account of their result which istotally geometric. In fact, we return to the Bestvina–Bradyconstruction and simplify their argument considerably by bringinga more general notion of links into play. At the end of thefirst section we re-prove their main result. By re-computingthe full -invariants, we show in the second section that thesimplification even adds some power to the method. The criterionwe give provides new insight on the geometric nature of the‘n-domination’ condition employed in [6].     

The Normal Holonomy Group of Kahler Submanifolds

We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a Kähler–Einsteinsubmanifold of CPⁿ has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a Kähler–Einstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full Kähler–Einsteinsubmanifold of CPⁿ is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary).     

Complete Manifolds with Sectional Curvature Bounded Below and Large Volume Growth

This paper provides a proof that an n-dimensional complete openRiemannian manifold M with sectional curvature K_M –1is diffeomorphic to a Euclidean n-space Rⁿ if the volume growthof geodesic balls in M is close to that of the balls in an n-dimensionalhyperbolic space Hⁿ(–1) of sectional curvature –1.     

On time regularity and related conditions for power-bounded operators

Let T be a bounded linear operator in a complex Banach space.Our main result gives various characterizations of the condition:T is power-bounded and an estimate ||(I – T)Tⁿ || cn^–1/2 holds for all positive integers n. In particular, this conditionholds if and only if T = β S + (1 – β)I, forsome β (0, 1) and some power-bounded operator S; or ifand only if T is power-bounded and the discrete semigroup (Tⁿ)is dominated by the continuous semigroup (e^{– t(I –T)})_{t 0} in a natural sense. As a consequence of our main results,for 1/2 < 1 we characterize the condition that T is power-boundedand ||(I – T)Tⁿ || c n^– for all n, in terms ofestimates on the semigroup e^{–t(I – T)}.     

A Topological Criterion for the Existence of Half-Bound States

The following theorem is proved: if (M⁴ⁿ⁺¹,g) is a completeRiemannian manifold and M is an oriented hypersurface partitioningM and with non-zero signature, then the spectrum of the Hodge–deRhamLaplacian is [0,]. This result is obtained by a new Callias-typeindex. This new formula links half-bound harmonic forms (thatis, nearly L² but not in L²) with the signature of .     

A Mod Two Analogue of a Conjecture of Cooke

The mod two cohomology of the three connective covering of S³has the form F₂[X_2n] E(Sq¹X_2n) where x_2n is in degree 2n and n = 2. If F denotes the homotopytheoretic fibre of the map S³ B²S¹ of degree 2, then the mod2 cohomology of F is also of the same form for n = 1. Notice(cf. Section 7 of the present paper) that the existence of spaceswhose cohomology has this form for high values of n would immediatelyprovide Arf invariant elements in the stable stem. Hence, itis worthwhile to determine for what values of n the above algebracan be realized as the mod2 cohomology of some space. The purposeof this paper is to construct a further example of a space withsuch a cohomology algebra for n = 4 and to show that no othervalues of n are admissible. More precisely, we prove the following.     

Strong Jordan Separation and Applications to Rigidity

We prove that simple, thick hyperbolic P-manifolds of dimensionat least three exhibit Mostow rigidity. We also prove a quasi-isometryrigidity result for the fundamental groups of simple, thickhyperbolic P-manifolds of dimension at least three. The keytool in the proof of these rigidity results is a strong formof the Jordan separation theorem, for maps from Sⁿ Sⁿ⁺¹ whichare not necessarily injective.     

Boundary Accessibility of a Domain Quasiconformally Equivalent to a Ball

A boundary point of a domain D in Rⁿ is said to be broadly accessibleif it ‘almost lies’ on the boundary of a round ballcontained in D. If f is a quasiconformal mapping of the unitball Bⁿ onto D, then it is shown that broadly accessible boundarypoints on D correspond under f to a set of full measure on Bⁿ.2000 Mathematics Subject Classification 30C65.     

Matheron's Conjecture for the Covariogram Problem

The covariogram of a convex body K provides the volumes of theintersections of K with all its possible translates. Matheronconjectured in 1986 that this information determines K amongall convex bodies, up to certain known ambiguities. It is provedthat this is the case if K R² is not C¹, or it is not strictlyconvex, or its boundary contains two arbitrarily small C² openportions ‘on opposite sides’. Examples are alsoconstructed that show that this conjecture is false in Rⁿ forany n 4.     

Finite-Element Approximation of Elliptic Equations with a Neumann or Robin Condition on a Curved Boundary

This paper considers a finite-element approximation of a second-orderself adjoint elliptic equation in a region Rⁿ (with n=2 or 3)having a curved boundary on which a Neumann or Robin conditionis prescribed. If the finite-element space defined over , a union of elements, has approximation power h^kin the L² norm, and if the region of integration is approximatedby ^h with dist (, ^h)Ch^k, then it is shown that one retains optimalrates of convergence for the error in the H¹ and L² norms, whetherQ^h is fitted or unfitted , provided that the numerical integration scheme has sufficientaccuracy.     

On Hardy, BMO and Lipschitz spaces of invariantly harmonic functions in the unit ball

The invariantly harmonic functions in the unit ball Bⁿ in Cⁿare those annihilated by the Bergman Laplacian . The Poisson-Szegökernel P(z,) solves the Dirichlet problem for : if f C(Sⁿ),the Poisson-Szegö transform of f, where d is the normalized Lebesgue measure on Sⁿ,is the unique invariantly harmonic function u in Bⁿ, continuousup to the boundary, such that u=f on Sⁿ. The Poisson-Szegötransform establishes, loosely speaking, a one-to-one correspondencebetween function theory in Sⁿ and invariantly harmonic functiontheory in Bⁿ. When n 2, it is natural to consider on Sⁿ functionspaces related to its natural non-isotropic metric, for theseare the spaces arising from complex analysis. In the paper,different characterizations of such spaces of smooth functionsare given in terms of their invariantly harmonic extensions,using maximal functions and area integrals, as in the correspondingEuclidean theory. Particular attention is given to characterizationin terms of purely radial or purely tangential derivatives.The smoothness is measured in two different scales: that ofSobolev spaces and that of Lipschitz spaces, including BMO andBesov spaces. 1991 Mathematics Subject Classification: 32A35,32A37, 32M15, 42B25.     

Diophantine approximation on non-degenerate curves with non-monotonic error function

It is shown that a non-degenerate curve in ⁿ satisfies a convergentGroshev theorem with a non-monotonic error function. In otherwords it is shown that if a volume sum converges the set ofpoints lying on the curve which satisfy a Diophantine conditionhas Lebesgue measure zero.