Showing papers in "Commentarii Mathematici Helvetici in 1998"

A classification of solutions of a conformally invariant fourth order equation in Rn

Abstract: In this paper, we consider the following conformally invariant equations of fourth order¶ $ \cases {\Delta^2 u = 6 e^{4u} &in $\bf {R}^4,$ \cr e^{4u} \in L^1(\bf {R}^4),\cr}$ (1)¶and¶ $ \cases {\Delta^2 u = u^{n+4 \over n-4}, \cr u>0 & in $ {\bf R}^n $ \qquad for $ n \ge5 $, \cr} $ (2) where $ \Delta^2 $ denotes the biharmonic operator in R n . By employing the method of moving planes, we are able to prove that all positive solutions of (2) are arised from the smooth conformal metrics on S n by the stereograph projection. For equation (1), we prove a necessary and sufficient condition for solutions obtained from the smooth conformal metrics on S 4 .

The behaviour at infinity of the Bruhat decomposition

Abstract: For a connected reductive group G and a Borel subgroup B, we study the closures of double classes BgB in a \( (G \times G) \)-equivariant "regular" compactification of G. We show that these closures \( \overline {BgB} \) intersect properly all \( (G \times G) \)-orbits, with multiplicity one, and we describe the intersections. Moreover, we show that almost all \( \overline {BgB} \) are singular in codimension two exactly. We deduce this from more general results on B-orbits in a spherical homogeneous space G/H; they lead to formulas for homology classes of H-orbit closures in G/B, in terms of Schubert cycles.

Higher generation subgroup sets and the Sigma-invariants of graph groups

Abstract: We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character 2 Hom(G; ) represents a point in the homotopical or homological -invariants of the group G.L et Gbe a nite simplicial graph,b the flag complex induced byG ,a nd GGthe graph group, or 'right angled Artin group', dened byG .W e use our result on n-generating subgroup sets to describe the homotopical and homological -invariants of GG in terms of the topology of subcomplexes of b. In particular, this work determines the niteness properties of kernels of maps from graph groups to abelian groups. This is the rst complete computation of the -invariants for a family of groups whose higher invariants are not determined | either implicitly or explicitly | by 1 .

$\pi_1$-injective surfaces in graph manifolds

Abstract: A criterion is given for an immersed horizontal $\pi_1$ -injective surface in a graph manifold to be separable. Examples are constructed of such surfaces, which are not separable and do not satisfy the k-plane property, for any k. It is shown that the simple loop conjecture holds in graph manifolds and that any graph manifold with boundary has an immersed horizontal surface.

Fano contact manifolds and nilpotent orbits

Abstract: A contact structure on a complex manifold M is a corank 1 subbundle F of TM such that the bilinear form on F with values in the quotient line bundle L = TM/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In this paper we consider the case where M is a Fano manifold; this implies that L is ample.¶If ${\frak g}$ is a simple Lie algebra, the unique closed orbit in ${\bold P}({\frak g})$ (for the adjoint action) is a Fano contact manifold; it is conjectured that every Fano contact manifold is obtained in this way. A positive answer would imply an analogous result for compact quaternion-Kahler manifolds with positive scalar curvature, a longstanding question in Riemannian geometry.¶In this paper we solve the conjecture under the additional assumptions that the group of contact automorphisms of M is reductive, and that the image of the rational map M $--\rightarrow$ P(H 0(M, L)*) sociated to L has maximum dimension. The proof relies on the properties of the nilpotent orbits in a semi-simple Lie algebra, in particular on the work of R. Brylinski and B. Kostant.

Schubert polynomials and Bott-Samelson varieties

Abstract: Schubert polynomials generalize Schur polynomials, but it is not clear how to generalize several classical formulas: the Weyl character formula, the Demazure character formula, and the generating series of semistandard tableaux. We produce these missing formulas and obtain several surprising expressions for Schubert polynomials.¶The above results arise naturally from a new geometric model of Schubert polynomials in terms of Bott-Samelson varieties. Our analysis includes a new, explicit construction for a Bott-Samelson variety Z as the closure of a B-orbit in a product of flag varieties. This construction works for an arbitrary reductive group G, and for G = GL(n) it realizes Z as the representations of a certain partially ordered set.¶This poset unifies several well-known combinatorial structures: generalized Young diagrams with their associated Schur modules; reduced decompositions of permutations; and the chamber sets of Berenstein-Fomin-Zelevinsky, which are crucial in the combinatorics of canonical bases and matrix factorizations. On the other hand, our embedding of Z gives an elementary construction of its coordinate ring, and allows us to specify a basis indexed by tableaux.

Approximations of stable actions on $ \Bbb {R} $-trees

Abstract: This article shows how to approximate a stable action of a finitely presented group on an \( \Bbb {R} \)-tree by a simplicial one while keeping control over arc stabilizers. For instance, every small action of a hyperbolic group on an \( \Bbb {R} \)-tree can be approximated by a small action of the same group on a simplicial tree. The techniques we use highly rely on Rips's study of stable actions on \( \Bbb {R} \)-trees and on the dynamical study of exotic components by D. Gaboriau.

Groups acting on finite dimensional spaces with finite stabilizers

Abstract: It is shown that every H \( \frak {F} \)-group G of type \( \rm{FP}_\infty \) admits a finite dimensional G-CW-complex X with finite stabilizers and with the additional property that for each finite subgroup H, the fixed point subspace X H is contractible. This establishes conjecture (5.1.2) of [9]. The construction of X involves joining a family of spaces parametrized by the poset of non-trivial finite subgroups of G and ultimately relies on the theorem of Cornick and Kropholler that if M is a \( \Bbb {Z} G \)-module which is projective as a \( \Bbb {Z} H \)-module for all finite \( H \le G \) then M has finite projective dimension.

The double Coxeter arrangement

Abstract: Let V be Euclidean space. Let $ W \subset {\bf G L}(V) $ be a finite irreducible reflection group. Let $ \cal {A} $ be the corresponding Coxeter arrangement. Let S be the algebra of polynomial functions on V. For $ H \in \cal {A} $ choose $ \alpha_H \in V^* $ such that $ H = {\rm ker}(\alpha_H) $ . The arrangement $ {\cal A} $ is known to be free: the derivation module $ D({\cal A}) = \{ \theta \in {\rm Der}_S ~|~ \theta(\alpha_H) \in S \alpha_H \} $ is a free S-module with generators of degrees equal to the exponents of W. In this paper we prove an analogous theorem for the submodule $ E({\cal A}) $ of $ D({\cal A})$ defined by $ E({\cal A}) = \{\theta \in {\rm Der}_S ~|~ \theta(\alpha_H) \in S \alpha_H^2\} $ . The degrees of the basis elements are all equal to the Coxeter number. The module $ E({\cal A}) $ may be considered a deformation of the derivation module for the Shi arrangement, which is conjectured to be free. The proof is by explicit construction using a derivation introduced by K. Saito in his theory of flat generators.

Deforming abelian SU(2)-representations of knot groups

Abstract: The aim of this paper is to generalize at heorem of C. D. Frohman and E. P. Klassen ((FK91)) concerning deformations of abelian SU(2)-representations of knot groups into non- abelian representations. The proof of our main theorem makes use of a generalization of a result of X.-S. Lin ((Lin92)) which should be interesting in itself.

The structure of branching in Anosov flows of 3-manifolds

Abstract: In this article we study the topology of Anosov flows in 3-manifolds. Specifically we consider the lifts to the universal cover of the stable and unstable foliations and analyze the leaf spaces of these foliations. We completely determine the structure of the non Hausdorff points in these leaf spaces. There are many consequences: (1) when the leaf spaces are non Hausdorff, there are closed orbits in the manifold which are freely homotopic, (2) suspension Anosov flows are, up to topological conjugacy, the only Anosov flows without free homotopies between closed orbits, (3) when there are infinitely many stable leaves (in the universal cover) which are non separated from each other, then we produce a torus in the manifold which is transverse to the Anosov flow and therefore is incompressible, (4) we produce non Hausdorff examples in hyperbolic manifolds and derive important properties of the limit sets of the stable/unstable leaves in the universal cover.

Cogroups in algebras over an operad are free algebras

Abstract: Let \(\cal P\) be an operad defined over a field of characteristic zero. Let R be a cogroup in the category of complete \({\cal P}\)-algebras. In this article, we show that R is necessarily the completion of a free \({\cal P}\)-algebra. We also handle the case of cogroups in connected graded algebras over an operad, and the case of groups in connected graded coalgebras over an operad.

S.A.G.B.I. bases for rings of formal modular seminvariants

Abstract: We use the theory of S.A.G.B.I. bases to construct a generating set for the ring of invariants for the four and five dimensional indecomposable modular representations of a cyclic group of prime order. We observe that for the four dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–3, and for the five dimensional representation the ring of invariants is generated in degrees less than or equal to 2p–2.

p-universal spaces and rational homotopy types

Abstract: We prove that the p-universality of a space does not depend on a prime p but only on its rational homotopy type. The minimal model of such a rational homotopy type is characterized by the existence of the trivial endomorphism in the closure of its automorphism group.

Continuously many quasiisometry classes of 2-generator groups

Abstract: We construct continuously many quasiisometry classes of torsion-free 2-generator small cancellation groups.

Stable constant mean curvature hypersurfaces in some Riemannian manifolds

Abstract: We determine all stable constant mean curvature hypersurfaces in a wide class of complete Riemannian manifolds having a foliation whose leaves are umbilical hypersurfaces. Among the consequences of this analysis we obtain all the stable constant mean curvature hypersurfaces in many nonsimply connected hyperbolic space forms.

Degenerations for representations of extended Dynkin quivers

Abstract: Let A be the path algebra of a quiver of extended Dynkin type $ \tilde {\Bbb {A}}_n, \tilde {\Bbb {D}}_n, \tilde {\Bbb {E}}_6, \tilde {\Bbb {E}}_7 $ or $ \tilde {\Bbb {E}}_8 $ . We show that a finite dimensional A-module M degenerates to another A-module N if and only if there are short exact sequences $ 0 \to U_i \to M_i \to V_i \to 0 $ of A-modules such that $ M = M_1 $ , $ M_{i+1} = U_i \oplus V_i $ for $ 1 \leq i \leq s $ and $ N = M_{s+1} $ are true for some natural number s.

A vanishing theorem for modular symbols on locally symmetric spaces

Abstract: A modular symbol is the fundamental class of a totally geodesic submanifold 0 nG 0 =K 0 embedded in a locally Riemannian symmetric space nG=K, which is dened by a subsymmetric space G 0 =K 0 ,! G=K. In this paper, we consider the modular symbol dened by a semisimple symmetric pair (G;G 0 ), and prove a vanishing theorem with respect to the -component (2b) in the Matsushima-Murakami formula based on the discretely decomposable theorem of the re- striction jG0 . In particular, we determine explicitly the middle Hodge components of certain totally real modular symbols on the locally Hermitian symmetric spaces of type IV.

The spectrum of the Chern subring

Abstract: For certain subrings of the mod-p cohomology ring of a compact Lie group, we give a description of the prime ideal spectrum, analogous to Quillen's description of the spectrum of the whole ring. Examples of such subrings include the Chern subring (the subring generated by Chern classes of all unitary representations), and for finite groups the subring generated by Chern classes of representations realizable over any specified field. As a corollary, we deduce that the inclusion of the Chern subring in the cohomology ring is an F-isomorphism for a compact Lie group G if and only if the following condition holds: For any homomorphism f between elementary abelian p-subgroups of G such that f(v) is always conjugate to v, there is an element g of G such that f is equal to conjugation by g.

Un théorème de rigidité différentielle

Abstract: Nous demontrons dans cet article le resultat de rigidite suivant, concernant le volume minimal d'une variete lisse fermee de dimension \( \ge 3 \).¶Theoreme: soient N et M deux varietes lisses, fermees, orientees de meme dimension \( n \ge 3 \). On suppose que M est munie d'une metrique hyperbolique g 0. Si \( f : N \rightarrow M \) est une application continue de degre non nul telle que \( {\rm Minvol} (N) = | \deg f | {\rm vol}_{g_0} (M) \), alors N est une variete hyperbolique et f est homotope a un revetement riemannien. La preuve repose sur l'utilisation de theoremes de convergence riemannienne a la Gromov [GLP], et sur l'adaptation de la construction de Besson, Courtois, Gallot [BCG].¶ L'une des applications interessantes est que le volume minimal n'est pas un invariant du type topologique de la variete, mais de la structure differentielle. Il n'est pas non plus additif par somme connexe.

The orbit space of the p-subgroup complex is contractible

Abstract: We show that the quotient space of the p-subgroup complex of a finite group by the action of the group is contractible. This was conjectured by Webb.

Curvature of curvilinear 4-webs and pencils of one forms: Variation on a theorem of Poincaré, Mayrhofer and Reidemeister

Abstract: A curvilinear d-web W = (F1 , . . . , Fd) is a configuration of d curvilinear foliations Fi on a surface. When d = 3, Bott connections of the normal bundles of Fi extend naturally to equal affine connection, which is called Chern connection. For 3 < d, this is the case if and only if the modulus of tangents to the leaves of Fi at a point is constant. A d-web is associative if the modulus is constant and weakly associative if Chern connections of all 3-subwebs have equal curvature form. We give a geometric interpretation of the curvature form in terms of fake billiard in §2, and prove that a weakly associative d-web is associative if Chern connections of triples of the members are non flat, and then the foliations are defined by members of a pencil (projective linear family of dim 1) of 1-forms. This result completes the classification of weakly associative 4-webs initiated by Poincare, Mayrhofer and Reidemeister for the flat case. In §4, we generalize the result for n + 2-webs of n-spaces.

Regularity properties of H-graphs

Abstract: It is proved that if H(u) is non-decreasing and if \( H (-\infty) eq H (+\infty) \), then if u (x) describes a graph over a disk BR (0), with (upward oriented) mean curvature H(u), there is a bound on the gradient \( | Du(0) | \) that depends only on R, on u (0), and on the particular function H (u). As a consequence a form of Harnack's inequality is obtained, in which no positivity hypothesis appears. The results are qualitatively best possible, in the senses a) that they are false if H is constant, and b) the dependences indicated are essential.¶The demonstrations are based on an existence theorem for a nonlinear boundary problem with singular data, which is of independent interest.

The representation ring of a compact Lie group revisited

Abstract: We describe a new construction of the induction homomorphism for representation rings of compact Lie groups: a homomorphism first defined by Graeme Segal. The idea is to first define the induction homomorphism for class functions, and then show that this map sends characters to characters. This requires a detection theorem - a class function of G is a character if its restrictions to certain subgroups of G are characters - which in turn requires a review of the representation theory for nonconnected compact Lie groups.

Applications harmoniques, applications pluriharmoniques et existence de 2-formes parallèles non nulles

Abstract: In this paper we study harmonic maps from a compact riemannian manifold equiped with a non trivial parallel 2-form, to a Kahler manifold of strongly negative curvature tensor or a riemannian manifold of strictly negative complex sectional curvature. In a first part we set up some rigidity results of Siu type. Then we obtain an upper bound for the rank of such maps in terms of the rank of the 2-form and deduce some vanishing theorems.

Dérivée des petites valeurs propres des surfaces de Riemann

Abstract: On estime la derivee des petites valeurs propres du Laplacien sur une famille de surfaces de Riemann. Ces valeurs propres sont considerees comme des fonctions sur l'espace de Teichmuller, et l'estimation des derivees peut s'exprimer dans ce contexte.

An example of an immersed complete genus one minimal surface in $ \Bbb {R}^3 $ with two convex ends

Abstract: We prove the existence of a compact genus one immersed minimal surface M, whose boundary is the union of two immersed locally convex curves lying in parallel planes. M is a part of a complete minimal surface with two finite total curvature ends.

Minimal orbits close to periodic frequencies

Abstract: Let \({\cal L}(Q,\dot Q)={1\over2}\vert\dot{Q}\vert^2+h(Q,\dot Q)\) with h analytic of small norm. The problem of Arnold's diffusion consists in finding conditions on h which guarantee the existence of orbits Q of \({\cal L}\) with \(\dot Q\) connecting two arbitrary points of frequency space. Recently, J. N. Mather has found a sufficient condition for Arnold's diffusion; this condition is not read on h itself, but on the set of all action-minimizing orbits of \({\cal L}\). In this paper we try to characterize those action-minimizing orbits whose mean frequency is close to periodic.