On the existence of closed timelike geodesics in compact spacetimes. II

 The purpose of this paper is on the one hand to extend and generalize, in terms of Clifford translations, some results in a previous paper (Math. Z. 239 (2002), 277–291) concerning the existence of closed timelike geodesics in compact spacetimes, and on the other hand to prove that a compact flat spacetime (M, g) contains a closed timelike geodesic if and only if the fundamental group π₁(M) contains a non-trivial timelike translation. Received: 22 January 2002; in final form: 12 August 2002 / Published online: 16 May 2003 Mathematics Subject Classification (2000): 53C50, 53C22.     

Congruence of minimal surfaces and higher fundamental forms

 We show that minimal surfaces in space forms are determined, up to ambient isometries, by the induced metric and certain invariants which are defined in terms of the higher fundamental forms and the complex structure. Received: 6 May 2002 / Revised version: 5 July 2002 Current address: Institut für Mathematik, Universit?t Augsburg, Universit?tsstrasse 12, D-86135 Augsburg, Germany. e-mail: Theodoros.Vlachos@Math.Uni-Augsburg.De The author was supported by the Alexander von Humboldt Foundation Mathematics Subject Classification (2000): 53A10, 53C42 Acknowledgements. This work was written during the author's stay at the University of Augsburg as a research fellow of the Alexander von Humboldt Foundation. I wish to express my gratitude to the Alexander von Humboldt Foundation for its generous support. Moreover, I wish to thank Professor J.H. Eschenburg for many fruitful and stimulating conversations.     

The electric field integral equation on Lipschitz screens: definitions and numerical approximation

Summary.  We use the integral equation approach to study electromagnetic scattering by perfectly conducting (non-orientable) Lipschitz screens. The well-posedness of the electric field integral equation is derived. The Galerkin method for this problem is analysed in a general setting and optimal error bounds are proved for conforming finite elements in natural norms. Received September 7, 2001 / Revised version received May 15, 2002 Published online October 29, 2002 Mathematics Subject Classification (1991): 65R20 Work of the second author was sponsored by Thales Airborne Systems.     

A sharp isoperimetric inequality for rank one symmetric spaces

 We prove an isoperimetric inequality for compact, regular domains in rank one symmetric spaces, which is sharp for geodesic balls. Besides volume and area of a given domain, some weak information about the second fundamental form of its boundary is involved. Received: 2 September 2002 / Revised version: 10 December 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 53C35, 52A40, 51M25     

Symbolic coding for the geodesic flow associated to a word hyperbolic group

 Let Γ be a word hyperbolic group M. Gromov has constructed a compact space equipped with a flow which is defined up to orbit-equivalence and which is called the geodesic flow of Γ. In the special case where Γ is the fundamental group of a Riemannian manifold of negative sectional curvature, is the unit tangent bundle of the manifold equipped with the usual geodesic flow. In this paper, we construct, for every hyperbolic group Γ, a subshift of finite type and a continuous map from the suspension of this subshift onto , which is uniformly bounded-to-one and which sends each orbit of the suspension flow onto an orbit of the geodesic flow. Received: 25 January 2002 / Revised version: 20 August 2002 Mathematics Subject Classification (2000): 20F67, 20F65, 20F69, 53C23, 53C21, 37D40, 37B10, 54H20     

<Emphasis Type="Italic">Q</Emphasis>-measures on <Emphasis Type="Italic">Q</Emphasis><Subscript><Emphasis Type="Italic">κ</Emphasis></Subscript>λ

 We give a characterization of strongly compact cardinals in terms of Q _κ λ. We also prove that weakly normal Q-measures on Q _κ λ are ⊂κ-normal. Received: 29 September 2000 / Revised version: March 2002 Published online: 5 November 2002 This project is supported by the New Zealand Marsden Fund. The author wishes to thank the referee for numerous comments and suggestions which have been incorporated into this version of the paper.     

Polyhedral <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">2</Emphasis></Subscript>

 Using elementary graded automorphisms of polytopal algebras (essentially the coordinate rings of projective toric varieties) polyhedral versions of the group of elementary matrices and the Steinberg and Milnor groups are defined. They coincide with the usual K-theoretic groups in the special case when the polytope is a unit simplex and can be thought of as compact/polytopal substitutes for the tame automorphism groups of polynomial algebras. Relative to the classical case, many new aspects have to be taken into account. We describe these groups explicitly when the underlying polytope is 2-dimensional. Already this low-dimensional case provides interesting classes of groups. Received: 13 December 2001 / Revised version: 24 June 2002 The second author was supported by the Deutsche Forschungsgemeinschaft, INTAS grant 99-00817 and TMR grant ERB FMRX CT-97-0107 Mathematics Subject Classification (2000): 14L27, 14M25, 19C09, 52B20     

A note on radial graphs with constant mean curvature

 Let Ω be a smooth domain on the unit sphere 𝕊ⁿ whose closure is contained in an open hemisphere and denote by ℋ the mean curvature of ∂Ω as a submanifold of Ω with respect to the inward unit normal. It is proved that for each real number H that satisfies inf ℋ > − H ≥ 0, there exists a unique radial graph on Ω bounded by ∂Ω with constant mean curvature H. The orientation on the graph is based on the normal that points on the opposite side as the radius vector. Received: 5 June 2001 / Revised version: 9 April 2002 Research partially supported by a DGICYT Grant No. BFM2001-2967. Mathematics Subject Classification (2000): 53A10, 53C42, 49Q05, 49Q10     

An extension theorem in symplectic geometry

 We extend the ``Extension after Restriction Principle' for symplectic embeddings of bounded starlike domains to a large class of symplectic embeddings of unbounded starlike domains. Received: 21 January 2002 / Revised version: 5 July 2002 Mathematics Subject Classification (2000): Primary 53D35, Secondary 54C20     

Topological restrictions for circle actions and harmonic morphisms

 Let M ^m be a compact oriented smooth manifold admitting a smooth circle action with isolated fixed points which are isolated as singularities as well. Then all the Pontryagin numbers of M ^m are zero and its Euler number is nonnegative and even. In particular, M ^m has signature zero. We apply this to obtain non-existence of harmonic morphisms with one-dimensional fibres from various domains, and a classification of harmonic morphisms from certain 4-manifolds. Received: 16 May 2002 Published online: 14 February 2003 Mathematics Subject Classification (2000): 58E20, 53C43, 57R20.     

Variational conditions with smooth constraints: structure and analysis

 This is an expository paper about the analysis of variational conditions over sets defined in finite-dimensional spaces by fairly smooth functions satisfying a constraint qualification. The primary focus is on results that can provide quantitative and computable sensitivity information for particular instances of the problems under study, and our objective is to give a personal view of the state of current knowledge in this area and of gaps in that knowledge that require future work. The writing style is informal, in keeping with the objective of focusing the reader's attention on the basic concepts and the relationships between them, rather than on details of the particular results themselves. Received: December 1, 2002 / Accepted: April 25, 2003 Published online: May 28, 2003 Key words. variational condition – variational inequality – complementarity – sensitivity – stability – nondegeneracy Mathematics Subject Classification (2000): Primary: 90C31. Secondary: 47J20, 49J40, 49J53, 90C33     

On a theorem of Wiener

 Wiener has shown that an integrable function on the circle T which is square integrable near the identity and has nonnegative Fourier transform, is square integrable on all of T. In the last 30 years this has been extended by the work of various authors step by step. The latest result states that, in a suitable reformulation, Wiener's theorem with ``p-integrable' in place of ``square integrable' holds for all even p and fails for all other p  (1, ∞) in the case of a general locally compact abelian group. We extend this to all IN-groups (locally compact groups with at least one invariant compact neighbourhood) and show that an extension to all locally compact groups is not possible: Wiener's theorem fails for all p < ∞ in the case of the ax + b-group. Received: 12 September 2000 Mathematics Subject Classification (2000): 43A35     

The stable forking conjecture and generic structures

 We prove that for any simple theory which is constructed via Fr?issé-Hrushovski method, if the forking independence is the same as the d-independence then the stable forking property holds. Received: 22 January 2001 / Published online: 19 December 2002 This article is part of the author's D-Phil thesis, written at the University of Oxford and supported by the Ministry of Higher Education of Iran. The author would like to thank the Institute for Studies in Theoretical Physics and Mathematics (IPM), Tehran, Iran, for its financial support whilst working on this article. Mathematics Subject Classification (2000): 03C45 Key words or phrases: Generic structures – Fr?issé-Hrushovski method – Predimension – Simple theories – Stable theories – Stable forking conjecture     

Extrinsic Killing spinors

 Under intrinsic and extrinsic curvature assumptions on a Riemannian spin manifold and its boundary, we show that there is an isomorphism between the restriction to the boundary of parallel spinors and extrinsic Killing spinors of non-negative Killing constant. As a corollary, we prove that a complete Ricci-flat spin manifold with mean-convex boundary isometric to a round sphere, is necessarily a flat disc. Received: 2 February 2002; in final form: 1 August 2002 / Published online: 1 April 2003 Mathematics Subject Classification (1991): 53C27, 53C40, 53C80, 58G25 The authors would like to thank Lars Andersson for helpful discussions and for bringing to our knowledge the information regarding Remark 4. We are also grateful to the referee for pointing out that Corollary 5 and Corollary 6 are only valid when the boundary is at least 2-dimensional. Research of S. Montiel is partially supported by a Spanish MCyT grant No. BFM2001-2967     

Homogeneous spaces with polar isotropy

 We consider homogeneous spaces G/K with G a simple compact Lie group, endowed with an arbitrary G-invariant Riemannian metric. We classify those spaces where the action of K on G/K is polar and show that such spaces are locally symmetric. Moreover we give a classification of pairs (G,K) with G compact semisimple such that K has polar linear isotropy representation. Received: 16 May 2002 / Revised version: 8 November 2002 Published online: 3 March 2003 Mathematics Subject Classification (2000): 53C35, 57S15     

A notion of rank in set theory without choice

 Starting from the definition of `amorphous set' in set theory without the axiom of choice, we propose a notion of rank (which will only make sense for, at most, the class of Dedekind finite sets), which is intended to be an analogue in this situation of Morley rank in model theory. Received: 22 September 2000 / Revised version: 14 May 2002 Published online: 19 December 2002 The research of the first author was supported by the SERC. Mathematics Subject Classification (2000): 03E25 Key words or phrases: Rank – Degree – Amorphous     

Cotangent bundles of 4-dimensional hypercomplex Lie groups

 We study geometrical structures on the cotangent bundle T ^* G of a Lie group G which are left-invariant with respect to the Lie group structure on T ^* G determined by a left-invariant affine structure ∇ on G. In particular, we investigate the existence of conformally hyper-K?hler metrics and hyper-K?hler with torsion (HKT) structures on the cotangent bundle of hypercomplex 4-dimensional Lie groups. By applying In?nü-Wigner contractions to compact semisimple Lie algebras we obtain non semisimple Lie algebras endowed with invariant HKT structures. Received: 4 February 2002 / Revised version: 20 August 2002 Research partially supported by MURST and GNSAGA (Indam) of Italy Mathematics Subject Classification (2000): 53C26, 22E25     

Hájek basic fuzzy logic and Łukasiewicz infinite-valued logic

 Using the theory of BL-algebras, it is shown that a propositional formula ϕ is derivable in Łukasiewicz infinite valued Logic if and only if its double negation ˜˜ϕ is derivable in Hájek Basic Fuzzy logic. If SBL is the extension of Basic Logic by the axiom (φ & (φ→˜φ)) → ψ, then ϕ is derivable in in classical logic if and only if ˜˜ ϕ is derivable in SBL. Axiomatic extensions of Basic Logic are in correspondence with subvarieties of the variety of BL-algebras. It is shown that the MV-algebra of regular elements of a free algebra in a subvariety of BL-algebras is free in the corresponding subvariety of MV-algebras, with the same number of free generators. Similar results are obtained for the generalized BL-algebras of dense elements of free BL-algebras. Received: 20 June 2001 / Published online: 2 September 2002 This paper was prepared while the first author was visiting the Universidad de Barcelona supported by INTERCAMPUS Program E.AL 2000. The second author was partially supported by Grants 2000SGR-0007 of D. G. R. of Generalitat de Catalunya and PB 97-0888 of D. G. I. C. Y. T. of Spain. Mathematics Subject classification (2000): 03B50, 03B52, 03G25, 06D35 Keywords or Phrases: Basic fuzzy logic – Łukasiewicz logic – BL-algebras – MV-algebras – Glivenko's theorem     

ℱ<Emphasis Type="Italic">K</Emphasis>-convex functions on metric spaces

 By an ℱK-convex function on a length metric space, we mean one that satisfies f ⁿ ≥ −Kf on all unitspeed geodesics. We show that natural ℱK-convex (-concave) functions occur in abundance on metric spaces of curvature bounded above (below) by K in the sense of Alexandrov. We prove Lipschitz extension and approximation theorems for ℱK-convex functions on CAT(K) spaces. Received: 10 May 2002 Mathematics Subject Classification (2000): 53C70, 52A41