Open Dense Subset Research Articles

On nonpositively curved Euclidean submanifolds: splitting results

In this article, we prove that a n–dimensional, non–positively curved Euclidean submanifold with codimension p and with minimal index of relative nullity \(\nu = n - 2p\) is (in an open dense subset) locally the product of p hypersurfaces.

General stable bundles arising as normal bundles of projective curves

Let X be a smooth projective curve of genus g ³2. For all integers r, d with r>0 let M(X;r,d) be the moduli scheme of rank r stable vector bundles on X with degree d. Here we prove that for all integers n,d with n³3 and d ³(5n-8)g + 2n2-5n+4 there is an open dense subset W of M(X; n-1, (n+1)d +2g-2) such that for every N I W there is a non-degenerate degree d embedding of X in Pn with N as normal bundle. For the proof we use reducible smoothable curves.

Genericity of the multibump dynamics for almost periodic Duffing-like systems

In this paper we consider ‘slowly’ oscillating perturbations of almost periodic Duffing-like systems, i.e. systems of the form ü = u − (a(t) + α(wt))W′(u), t ∈ ℝ, u ∈ ℝN, where W ∈ C2N(ℝN, ℝ) is superquadratic and a and α are positive and almost periodic. By variational methods, we prove that if w > 0 is small enough, then the system admits a multibump dynamics. As a consequence we get that the system ü = u − a(t)W′(u), t ∈ ℝ, u ∈ ℝN, admits multibump solutions whenever a belongs to an open dense subset of the set of positive almost periodic continuous functions.

Transformations pentagonales

We prove that every bimeasurable transformation v : X x XX x X, where X is a measured space satisfying the pentagon identity v 23 o v 13 o v 12 = v 12 o v 23 (and sufficiently regular) comes from a matched pair of locally compact groups (cf. [5], [8], [6], [2]), i.e. a locally compact group G and two closed subgroups, G 1 and G 2 such that the map (g 1,g 2)g 1g 2 is a homeomorphism from G 1 x G 2 onto a dense open subset of G.

On the Zeros of Solutions of Linear Differential Equations of the Second Order

Let u be a solution of the differential equation u″+Ru=0, where R is rational. Newton's method of finding the zeros of u consists of iterating the function f(z)=z−u(z)/u′(z). With suitable hypotheses on R and u, it is shown that the iterates of f converge on an open dense subset of the plane if they converge for the zeros of R. The proof is based on the iteration theory of meromorphic functions, and in particular on the result that, if the family of K-quasiconformal deformations of a meromorphic function f depends on only finitely many parameters, then every cycle of Baker domains of f contains a singularity of f−1. This result, together with classical results of Hille concerning the asymptotic behaviour of solutions of the above differential equations, is also used to study their value distribution. For example, it is shown that, if R is a rational function which satisfies R(z)∼amzm as z→∞ and has only k distinct zeros where k<(m+2)/2, then δ(0, u)⩽k/(m+2−k)<1.

Diagrammes de Dynkin et algèbres enveloppantes d'algèbres de lie semi-simples

Let g be a semi-simple complex Lie algebra, U = U( g ) its enveloping algebra, and A a minimal primitive factor of U, with central character χ. Under the assumption that χ is regular and integral, we prove that the Dynkin diagram of g is a Morita invariant of A. Further, a slight refinement implies that the flag variety of g is determined, within all generalized flag varieties, by its ring of differential operators. Then we derive the following consequences. First, if U( g ) ≅ U( g ′), for some Lie algebra g ′, then g ′ ≅ g . Second, any automorphism of U acts on the centre, and on some dense open subset of the primitive spectrum, as a diagram automorphism. We conjecture that this result holds true on the whole primitive spectrum, and give K-theoretic versions of the result and the conjecture. We also improve a key result of [1]. Finally, when χ is only assumed to be regular, we prove, using a result of Soergel, that the Weyl group of g is a Morita invariant of A.

Classification of varieties with canonical curve section via gaussian maps on canonical curves

The purpose of this article is to develop further a method to classify varieties X ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] having canonical curve section, using Gaussian map computations. In a previous article we applied these techniques to classify prime Fano threefolds, that is Fano threefolds whose Picard group is generated by the hyperplane bundle. In this article we extend this method and classify Fano threefolds of higher index and Mukai varieties , i.e., varieties of dimension four or more with canonical curve sections. First we determine when the Hilbert scheme [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] of such varieties X is nonempty. Moreover, in the case of Picard number one, we prove that [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] is irreducible and that the examples of Fano-Iskovskih and Mukai form a dense open subset of smooth points of [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="04i" /].

NEW POLARIZATIONS ON THE MODULI SPACES AND THE THURSTON COMPACTIFICATION OF TEICHMÜLLER SPACE

Given a foliation F with closed leaves and with certain kinds of singularities on an oriented closed surface Σ, we construct in this paper an isotropic foliation on ℳ(Σ), the moduli space of flat G-connections, for G any compact simple simply connected Lie-group. We describe the infinitesimal structure of this isotropic foliation in terms of the basic cohomology with twisted coefficients of F. For any pair (F, g), where g is a singular metric on Σ compatible with F, we construct a new polarization on the symplectic manifold ℳ′(Σ), the open dense subset of smooth points of ℳ(Σ). We construct a sequence of complex structures on Σ, such that the corresponding complex structures on ℳ′(Σ) converges to the polarization associated to (F, g). In particular we see that the Jeffrey–Weitzman polarization on the SU(2)-moduli space is the limit of a sequence of complex structures induced from a degenerating family of complex structures on Σ, which converges to a point in the Thurston boundary of Teichmüller space of Σ. As a corollary of the above constructions, we establish a certain discontinuiuty at the Thurston boundary of Teichmüller space for the map from Teichmüller space to the space of polarizations on ℳ′(Σ). For any reducible finite order diffeomorphism of the surface, our constuction produces an invariant polarization on the moduli space.

Some ultrabornological normed function spaces

A variety of normed function spaces, including the space E 0 (X) of continuous functions on the compact Hausdorff space X that are locally constant on a dense open subset of X, are shown to be ultrabornological.

On the Regularity of Semipermeable Surfaces in Control Theory with Application to the Optimal Exit-Time Problem (Part I)

In control theory, a semipermeable surface is an (in general nonsmooth) oriented surface that, on one hand, contains solutions (the so-called barrier solutions) of the controlled system and, on the other hand, may be crossed by the solutions of this system in only one direction. Without making any assumption on the regularity of the boundary of the semipermeable surface, we show that the barrier solutions contained in this semipermeable surface satisfy the Pontryagin principle, that this surface is a Lipschitz manifold, and that it is, locally, the graph of a semiconcave function. Applying these results to the optimal exit-time function from a given open set yields, without any controllability assumption at the boundary of the open set, that this function is semiconcave on an open dense subset of its domain.

On the Regularity of Semipermeable Surfaces in Control Theorywith Application to the Optimal Exit-Time Problem (Part II)

In control theory, a semipermeable surface is an (in general nonsmooth) oriented surface that, on one hand, contains solutions (the so-called barrier solutions) of the controlled system and, on the other hand, may be crossed by the solutions of this system in only one direction. Without making any assumption on the regularity of the boundary of the semipermeable surface, we show that the barrier solutions contained in this semipermeable surface satisfy the Pontryagin principle, that this surface is a Lipschitz manifold, and that it is, locally, the graph of a semiconcave function. Applying these results to the optimal exit-time function from a given open set yields, without any controllability assumption at the boundary of the open set, that this function is semiconcave on an open dense subset of its domain.

Transmission of continuity to the Bohr topology

For a locally compact Abelian (LCA) group G, let G + denote the group G endowed with its Bohr topology. With each piecewise affine map (defined below) α of G into another LCA group H, we show that there is associated a continuous map α + of G + into H + which coincides with α on a dense open subset of G +. We study when α + is a homeomorphism, provided that α has this property. These ideas are applied to investigate to what extent the group algebra of integrable functions on an LCA group G, L 1 ( G), characterizes the group.

Quantification de la dualité de Grassmann

Let V be a complex vector space of dimension n, %plane1D;53E; (resp. %plane1D;53E; *) the Grassmann manifold of p-dimensional (resp. (n — p)-dimensional) subspaces of V, and of Ω the relation of transversality in %plane1D;53E;*%plane1D;53E; *. We announced in [6] equivalences between derived categories of sheaves and of D-modules on %plane1D;53E; and %plane1D;53E; defined by the integral transforms associated to Ω. We show here that these transforms exchange the D-modules associated to the holomorphic line bundles on %plane1D;53E; and %plane1D;53E; *. This is equivalent to “quantizing” the underlying contact transformation between certain open dense subsets of the cotangent bundles. In the case p = 1, we recover already known results for the projective duality (see [1] and [5]).

D‐Affinity and Toric Varieties

1991 Mathematics Subject Classification 32C38.

Generic properties of singular trajectories

Let M be a σ-compact C∞ manifold of dimension d ≥ 3. Consider on M a single-input control system : x˙(t)=G0(x(t))+u(t)F1(x(t)), where F0, F1 are C∞ vector fields on M and the set of admissible controls U is the set of bounded measurable mappings u : [0Tu]↦R, Tu > 0. A singular trajectory is an output corresponding to a control such that the differential of the input-output mapping is not of maximal rank. In this article we show that for an open dense subset of the set of pairs of vector fields (F0, F1), endowed with the C∞-Whitney topology, all the singular trajectories are with minimal order and the corank of the singularity is one.

A Fatou-type theorem for functions associated with conformal densities on the boundary of a metric tree

A metric tree is a metric space such that between any two points there is a unique topological segment (i.e., a subset homeomorphic to an interval of the real line), and such that this segment is a geodesic (i.e., the isometric image of an interval). The theory of group actions on metric trees and of the structure of these trees on the boundary at infinity has many different aspects, and it has been made clear that this theory captures a lot of the interesting phenomena of hyperbolic geometry, or, to be more precise, of the theory of groups acting by isometrics on the hyperbolic space I/~ and on its sphere at infinity. One of the particular features, in the case of metric trees, is that the ideas and methods which are used are often of a combinatorial nature, and, probably, simpler and more intuitive than those of classical hyperbolic geometry. In particular, in [2] and [3], we give a series of criteria for the ergodicity of the geodesic flow associated with groups acting properly and isometrically on metric trees, which are discrete analogs of theorems due to Hopf and Sullivan in the case of hyperbolic manifolds. The invariant measures on these flows come from conformal densities on the boundary. This theory of conformal densities on the boundary was developed, in the case of hyperbolic manifolds, by Patterson and Sullivan (see [5] and [6]), and later, in the case of metric trees, in [11 . The result that we prove here concerns a Fatou-type behavior of certain natural functions associated with conformal densities on the boundary of a metric tree. A somehow more special result of this kind, in the case of conformal densities on the boundary of the hyperbolic n-space, is used by Sullivan in [6], in his proof for the probabilistic criterion of the ergodicity of the geodesic flow on a hyperbolic manifold (see also [7, Theorem 2.14]). T h e p r o o f of the corresponding result for metric trees, which we give in [31, involves a different kind of argument, although the Fatou-type behavior is also valid in that case, and, indeed, the aim of the present paper is to prove this result in the case of metric trees. Let us begin by describing the background. Let X be a metric tree which is complete and locally compact. We denote the distance between two points in X by Ix YlA geodesic ray (resp. a geodesic segment) is a map f from [0, oo[ (resp. from a closed interval) into X such that I f(r t ) f(r2)l = Irt r~l for every r, and r2. The boundary OX is the space of the ends of X. Every geodesic ray r defines a point in OX, which is denoted by r(oo). Let us also recall that between any two points x E X and r /E OX, there is a unique image of a geodesic ray, which is denoted by Ix, r/[. There is a natural topology on X U OX, which can easily be defined with the use of the ray structure from any given point in X, and for this topology, the space X U OX is compact and X lies in i t as an open dense subset. Better than that, X tO OX is, in fact, equipped with a natural class of metrics, and the induced class of metrics on OX, called visual metrics, is most useful. Let us recall its definition: For every point z E X, the visual metric I I~ on OX is defined by

CAT(0) 4-manifolds possessing a single tame point are Euclidean

The concepts of geometric and topological tame point are introduced for a space of nonpositive curvature. These concepts are applied to the characterization problem forCAT(0) 4-manifolds. It is shown that everyCAT(0)M 4 having a single (geometric or topological) tame point is homeomorphic toR 4. Davis and Januszkiewicz have recently constructedCAT(0)n-manifolds,M n withn ≥ 5 such that the set of tame points form a dense open subset ofM n , butM n ≠R n .

Symbolic calculus on the nilpotent orbits of SO 0(1,2)

The coadjoint conical orbits in so(1,2) ∗ ⋍ su(1,1) ∗ are the phase spaces of the zero mass particles on the two-dimensional (anti-)de Sitter space-time. It contains also, as an open dense subset, the phase space for massless particles on one-dimensional Minkowski space-time when one identifies the Poincaré group to a subgroup of the conformal group SO 0(2,2)⋍SO 0(1,2)×SO 0(1,2). On the other hand, the quantum representation associated to these systems is an indecomposable extension of the first term of the discrete series of representations of SO 0(1,2) ⋍ SU(1,1)/ Z 2. We present in this paper a symbol map linking this representation and this orbit. This calculus is invariant and behaves correctly in the classical limit. As a result we have obtained a conformally invariant symbolic calculus for massless particles on (anti-)de Sitter or Minkowski space-time in 1+1 dimension.

Double-periodic solutions of nonlinear Cauchy-Riemann equations

For a compact Kählerian manifold M we study the equation where f(z,·) is a vector field on M, sufficiently smooth in z, u. We define V as the infimum of areas of nontrivial holomorphic spheroids S2 ∪ M. So V = ∞ if M is aspherical and, for example, V = 1 if M = S2 (more generally, if M = BPn). Let ‖ f(z,u)‖u < √V where ‖ · ‖u stands for the normalized Kählerian metric. Then, first, the equation has at least |X (M)| contractible solutions (X (M) is the Euler characteristic) if f is taken from an open dense subset of the set of all admissible f and, second, for each admissible f it has at least one solution if X (M) ≠ 0. These estimates are sharp. © 1996 John Wiley & Sons, Inc.

On normality of the closure of a generic torus orbit inG∕P

In this paper we consider generic orbits for the action of a maximal torus T in a connected semisimple algebraic group G on the generalized flag variety G/P, where P is a parabolic subgroup of G containing T. The union of all generic T-orbits is an open dense (possibly proper, if P is not a Borel subgroup) subset of the intersection of the big cells in G/P. We prove that the closure of a generic Γ-orbit in G/P is a normal equivariant T-embedding (whose fan we explicitely describe). Moreover, the closures of any two generic Γ-orbits are isomorphic as equivariant T-embeddings.