Some global properties of static spacetimes

 We show that a static Lorentzian manifold satisfying a completeness assumption is geodesically connected. In particular, such condition is satisfied by all compact static manifolds; in the compact case, we also prove the existence of a closed geodesic in every free homotopy class determined by a finite number of deck transformations. Received: 28 July 2002 / Published online: 16 May 2003 Mathematics Subject Classification (1991): 53C22, 53C50, 58E10 The first and the second author are sponsored by the MIUR national project ``Metodi variazionali e topologici nello studio di fenomeni nonlineari'. The third author is partially sponsored by CNPq, Brazil.     

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     

Killing Forms on Quaternion-Kähler Manifolds

We show that every Killing p-form on a compact quaternion manifold has to be parallel for p≥ 2. Mathematics Subject Classifications (2000): Primary 53C55, 58J50.     

The Douglas problem for parametric double integrals

 Let be a parametric variational double integral and γ ⊂ ℝ ⁿ be a system of several distinct Jordan curves. We prove the existence of multiply connected, conformally parametrized minimizers of spanned in γ by solving the Douglas problem for parametric functionals on multiply connected schlicht domains. As a by-product we obtain a simple isoperimetric inequality for multiply connected -minimizers, and we discuss regularity results up to the boundary which follow from corresponding results for the Plateau problem. Received: 19 April 2002 Mathematics Subject Classification (2000): 49J45, 49Q10, 53A07, 53A10     

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.     

Some families of special Lagrangian tori

 We give an explicit proof of the local version of Bryant's result [1], stating that any 3-dimensional real-analytic Riemannian manifold can be isometrically embedded as a special Lagrangian submanifold in a Calabi-Yau manifold. We then refine the theorem proving that a certain class of real-analytic one-parameter families of metrics on a 3-torus can be isometrically embedded in a Calabi-Yau manifold as a one-parameter family of special Lagrangian submanifolds. Two applications of these results show how the geometry of the moduli space of 3-dimesional special Lagrangian submanifolds differs considerably from the 2-dimensional one. First of all, applying Bryant's theorem and a construction due to Calabi we show that nearby elements of the local moduli space of a special Lagrangian 3-torus can intersect themselves. Secondly, we use our examples of one-parameter families to show that in dimension three (and higher) the moduli space of special Lagrangian tori is not, in general, special Lagrangian in the sense of Hitchin [13]. Received: 18 December 2001 / Revised version: 31 January 2002 / Published online: 16 October 2002 Mathematics Subject Classification (2000): 53-XX, 53C38     

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     

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     

The contact boundary of a complex polynomial

 We define the contact boundary of a complex polynomial f : ℂ ⁿ → ℂ as the intersection of some generic fiber with a large sphere. We show that, up to contact isotopy, this does not depend on the choice of the fiber (provided it is generic) and is invariant under polynomial automorphism of ℂ ⁿ . We next prove that the formal homotopy class of this contact boundary is invariant in a large family of deformations of polynomials, which are not necessarily topologically trivial. Received: 15 November 2002 Published online: 20 March 2003 Mathematics Subject Classification (2000): 32S55, 53D15, 32S50     

Conformal vector fields on Kaehler manifolds

It is known that a conformal vector field on a compact Kaehler manifold is a Killing vector field. In this paper, we are interested in finding conditions under which a conformal vector field on a non-compact Kaehler manifold is Killing. First we prove that a harmonic analytic conformal vector field on a 2n-dimensional Kaehler manifold (n ≠ 2) of constant nonzero scalar curvature is Killing. It is also shown that on a 2n-dimensional Kaehler Einstein manifold (n > 1) an analytic conformal vector field is either Killing or else the Kaehler manifold is Ricci flat. In particular, it follows that on non-flat Kaehler Einstein manifolds of dimension greater than two, analytic conformal vector fields are Killing.     

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.     

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.     

Green function estimate for censored stable processes

 Sharp two-sided estimates for Green functions of censored α-stable process Y in a bounded C ^1,1 open set D are obtained, where α  (1, 2). It is shown that the Martin boundary and minimal Martin boundary of Y can all be identified with the Euclidean boundary of D. Sharp two-sided estimates for the Martin kernel of Y are also derived. Received: 27 January 2002 / Revised version: 10 June 2002 / Published online: 24 October 2002 This research is supported in part by NSF Grant DMS-0071486. Mathematics Subject Classification (2002): Primary: 60J45, 31C35; Secondary: 60G52, 31C15 Keywords or phrases: Censored stable process – Green function – Capacity – Martin boundary – Martin kernel – Harmonic function     

Singular Monge-Ampère foliations

 This paper generalizes results of Lempert and Sz?ke on the structure of the singular set of a solution of the homogeneous Monge-Ampère equation on a Stein manifold. Their a priori assumption that the singular set has maximum dimension is shown to be a consequence of regularity of the solution. In addition, their requirement that the square of the solution be C ³ everywhere is replaced by a smoothness condition on the blowup of the singular set. Under these conditions, the singular set is shown to inherit a Finsler metric, which in the real analytic case uniquely determines the solution of the Monge-Ampère equation. These results are proved using techniques from contact geometry. Received: 6 April 2001 / Published online: 2 December 2002 Mathematics Subject Classification (2000): 53C56, 32F, 53C60     

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     

Hydrodynamic limit for ∇φ interface model on a wall

 We consider random evolution of an interface on a hard wall under periodic boundary conditions. The dynamics are governed by a system of stochastic differential equations of Skorohod type, which is Langevin equation associated with massless Hamiltonian added a strong repelling force for the interface to stay over the wall. We study its macroscopic behavior under a suitable large scale space-time limit and derive a nonlinear partial differential equation, which describes the mean curvature motion except for some anisotropy effects, with reflection at the wall. Such equation is characterized by an evolutionary variational inequality. Received: 10 January 2002 / Revised version: 18 August 2002 / Published online: 15 April 2003 Mathematics Subject Classification (2000): 60K35, 82C24, 35K55, 35K85 Key words or phrases: Hydrodynamic limit – Effective interfaces – Hard wall – Skorohod's stochastic differential equation – Evolutionary variational inequality     

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     

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     

Vanishing of local homology

 We study the vanishing properties of local homology of complexes of modules without assuming that its homology is artinian. Using vanishing results for local homology and cohomology we prove new vanishing results for Ext- and Tor-modules. Received: 1 August 2002; in final form: 23 September 2002 / Published online: 16 May 2003 Mathematics Subject Classification (1991): 13C12, 13D07, 13D45.     

The Bando-Calabi-Futaki character and its lifting to a group character

 The Bando-Calabi-Futaki character of a compact K?hler manifold is an obstruction to the existence of K?hler metrics with constant scalar curvature, which is a generalization of the Futaki character of a Fano manifold. In this paper, we shall establish an interpretation of the Bando-Calabi-Futaki character as a secondary characteristic class. By using this interpretation, we shall also prove that the Bando-Calabi-Futaki character can be lifted to a group character. Received: 26 October 2000 / Revised version: 25 October 2001 / Published online: 16 October 2002 RID="★" ID="★" Partly supported by the Grant-in-Aid for Encouragement of Young Scientists (No. 09740047), The Ministry of Education, Science, Sports and Culture, Japan Mathematics Subject Classification (2000): 32Q20, 32Q15, 32M05, 57R30