Real Analytic Case Research Articles

Errata Corrige. Decomposition of CR-manifolds and splitting of CR-maps

We show the uniqueness of local and global decompositions of abstract CR-manifolds into direct products of irreducible factors, and a splitting property for their CR-diffeomorphisms into direct products with respect to these decompositions. The assumptions on the manifolds are finite non-degeneracy and finite-type on a dense subset. In the real-analytic case, these are the standard assumptions that appear in many other questions. In the smooth case, the assumptions cannot be weakened by replacing “dense” with “open” as is demonstrated by an example. An application to the cancellation problem is also given. The proof is based on the development of methods of [BER99b], [BRZ00], [KZ01] and the use of “approximate infinitesimal automorphisms” introduced in this paper.

Metrisability of two-dimensional projective structures

We carry out the programme of R. Liouville, Sur les invariants de certaines equations differentielles et sur leurs applications, to construct an explicit local obstruction to the existence of a Levi–Civita connection within a given projective structure $\Gamma$ on a surface. The obstruction is of order 5 in the components of a connection in a projective class. It can be expressed as a point invariant for a second order ODE whose integral curves are the geodesics of $\Gamma$ or as a weighted scalar projective invariant of the projective class. If the obstruction vanishes we find the sufficient conditions for the existence of a metric in the real analytic case. In the generic case they are expressed by the vanishing of two invariants of order 6 in the connection. In degenerate cases the sufficient obstruction is of order at most 8.

Clôture intégrale des idéaux et équisingularité

Clôture intégrale des idéaux et équisingularité

Finite jet determination of CR mappings

We prove the following finite jet determination result for CR mappings: Given a smooth generic submanifold M ⊂ C N , N ⩾ 2 , that is essentially finite and of finite type at each of its points, for every point p ∈ M there exists an integer ℓ p , depending upper-semicontinuously on p, such that for every smooth generic submanifold M ′ ⊂ C N of the same dimension as M, if h 1 , h 2 : ( M , p ) → M ′ are two germs of smooth finite CR mappings with the same ℓ p jet at p, then necessarily j p k h 1 = j p k h 2 for all positive integers k. In the hypersurface case, this result provides several new unique jet determination properties for holomorphic mappings at the boundary in the real-analytic case; in particular, it provides the finite jet determination of arbitrary real-analytic CR mappings between real-analytic hypersurfaces in C N of D'Angelo finite type. It also yields a new boundary version of H. Cartan's uniqueness theorem: if Ω , Ω ′ ⊂ C N are two bounded domains with smooth real-analytic boundary, then there exists an integer k, depending only on the boundary ∂ Ω, such that if H 1 , H 2 : Ω → Ω ′ are two proper holomorphic mappings extending smoothly up to ∂ Ω near some point p ∈ ∂ Ω and agreeing up to order k at p, then necessarily H 1 = H 2 .

Generalized Killing spinors in dimension 5

We study the intrinsic geometry of hypersurfaces in Calabi-Yau manifolds of real dimension 6 and, more generally, -structures on 5-manifolds defined by a generalized Killing spinor. We prove that in the real analytic case, such a 5-manifold can be isometrically embedded as a hypersurface in a Calabi-Yau manifold in a natural way. We classify nilmanifolds carrying invariant structures of this type, and present examples of the associated metrics with holonomy .

Generalized symplectic Cayley transforms and a higher order formula for the Conley–Zehnder index of symplectic paths

Using a generalized notion of symplectic Cayley transform in the symplectic group, we introduce a sequence of integer valued invariants (higher order signatures) associated with a degeneracy instant of a smooth path of symplectomorphisms. In the real analytic case, we give a formula for the Conley–Zehnder index in terms of the higher order signatures.

Curved flats, pluriharmonic maps and constant curvature immersions into pseudo-Riemannian space forms

We study two aspects of the loop group formulation for isometric immersions with flat normal bundle of space forms. The first aspect is to examine the loop group maps along different ranges of the loop parameter. This leads to various equivalences between global isometric immersion problems among different space forms and pseudo-Riemannian space forms. As a corollary, we obtain a non-immersibility theorem for spheres into certain pseudo-Riemannian spheres and hyperbolic spaces. The second aspect pursued is to clarify the relationship between the loop group formulation of isometric immersions of space forms and that of pluriharmonic maps into symmetric spaces. We show that the objects in the first class are, in the real analytic case, extended pluriharmonic maps into certain symmetric spaces which satisfy an extra reality condition along a totally real submanifold. We show how to construct such pluriharmonic maps for general symmetric spaces from curved flats, using a generalised DPW method.

Real and complex indices of vector fields on complete intersection curves with isolated singularity

If (V, 0) is an isolated complete intersection singularity and X a holomorphic vector field tangent to V, one can define an index of X, the so-called GSV index, which generalizes the Poincaré–Hopf index. We prove that the GSV index coincides with the dimension of a certain explicitly constructed vector space, if X is deformable in a certain sense and V is a curve. We also give a sufficient algebraic criterion for X to be deformable in this way. If one considers the real analytic case one can also define an index of X which is called the real GSV index. Under the condition that X has the deformation property, we prove a signature formula for the index generalizing the Eisenbud–Levine Theorem.

The Weyl–Lanczos equations and the Lanczos wave equation in four dimensions as systems in involution

The Weyl–Lanczos equations in four dimensions form a system in involution. We compute its Cartan characters explicitly and use Janet–Riquier theory to confirm the results in the case of all space–times with a diagonal metric tensor and for the plane wave limit of space–times. We write the Lanczos wave equation as an exterior differential system and, with assistance from Janet–Riquier theory, we compute its Cartan characters and find that it forms a system in involution. We compare these Cartan characters with those of the Weyl-Lanczos equations. All results hold for the real analytic case.

Stability and instability for Gevrey quasi-convex near-integrable Hamiltonian systems

Abstract. – We prove a theorem about the stability of action variables for Gevrey quasi-convex near-integrable Hamiltonian systems and construct in that context a system with an unstable orbit whose mean speed of drift allows us to check the optimality of the stability theorem.¶Our stability result generalizes those by Lochak-Neishtadt and Poschel, which give precise exponents of stability in the Nekhoroshev Theorem for the quasi-convex case, to the situation in which the Hamiltonian function is only assumed to belong to some Gevrey class instead of being real-analytic. For n degrees of freedom and Gevrey-α Hamiltonians, α ≥ 1, we prove that one can choose a = 1/2nα as an exponent for the time of stability and b = 1/2n as an exponent for the radius of confinement of the action variables, with refinements for the orbits which start close to a resonant surface (we thus recover the result for the real-analytic case by setting α = 1).¶On the other hand, for α > 1, the existence of compact-supported Gevrey functions allows us to exhibit for each n ≥ 3 a sequence of Hamiltonian systems with wandering points, whose limit is a quasi-convex integrable system, and where the speed of drift is characterized by the exponent 1/2(n−2)α. This exponent is optimal for the kind of wandering points we consider, inasmuch as the initial condition is located close to a doubly-resonant surface and the stability result holds with precisely that exponent for such an initial condition. We also discuss the relationship between our example of instability, which relies on a specific construction of a perturbation of a discrete integrable system, and Arnold’s mechanism of instability, whose main features (partially hyperbolic tori, heteroclinic connections) are indeed present in our system.

Singular Monge-Amp�re foliations

This paper generalizes results of Lempert and Szoke on the structure of the singular set of a solution of the homogeneous Monge-Ampere 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-Ampere equation. These results are proved using techniques from contact geometry.

Weak mixing for reparameterized linear flows on the torus

In this paper, we study the display of weak mixing by reparameterized linear flows on the torus \mathbb{T}^d, d\geq 2. We show that if the vector of the translation flow is Liouvillian (i.e. well approximated by rationals), then for a residual set of time change functions in the C^\infty topology, the reparameterized flow is weak mixing. If this is not the case, i.e. if the vector of the linear flow is Diophantine, it follows from a result of Kolmogorov on the two torus, and its generalization to any dimension by Herman, that any C^\infty reparameterization of the flow is C^\infty conjugate to a linear flow. More generally, in any given class of differentiability C^r for the time change function \phi, we give the optimal arithmetical condition on the vector of the translation flow that guarantees the existence of a residual set in the C^r topology of weak mixing reparameterizations. In the real analytic case, the optimal arithmetical condition for the generic display of weak mixing under time change is also given.As a consequence of our results on reparameterizations of Liouvillian linear flows, we obtain that an aperiodic smooth flow on the two-dimensional torus is in general weak mixing. We also deduce the existence on the torus of analytic diffeomorphisms that are rank one and weak mixing.

Quasiconformal variation of slit domains

We use quasiconformal variations to study Riemann mappings onto variable single slit domains when the slit is the tail of an appropriately smooth Jordan arc. In the real analytic case our results answer a question of Dieter Gaier and show that the function κ in Lowner's differential equation is real analytic.

Cantor sets in the line: scaling functions and the smoothness of the shift-map

Consider d disjoint closed subintervals of the unit interval and consider an orientation preserving expanding map which maps each of these subintervals to the whole unit interval. The set of points where all iterates of this expanding map are defined is a Cantor set. Associated with the construction of this Cantor set is the scaling function which records the infinitely deep geometry of this Cantor set. This scaling function is an invariant of conjugation. Dennis Sullivan posed the inverse problem: given a scaling function, determine the maximal possible smoothness of any expanding map which produces it. We solve this problem in the case of finite smoothness and in the real-analytic case.

Real Analytic Fibre Bundles: Some Results on Classification and Homotopies

AbstractThe classification problem for holomorphic fibre bundles over Stein spaces was solved by H. GRAUERT. Along the same lines, the real coherent analytic case was considered by A. TOGNOLI and V. ANCONA.In this paper we propose a different approach, based on classifying spaces, in order to study the previous problem for real analytic fibre bundles over C‐analytic subspaces of Rm.So, let X be a C‐analytic subspace of Rm and G a compact Lie group. The main result is a characterization of the real analytic G‐principal fibre bundles over X for which the analytic and topological equivalence coincide.Moreover, we prove that these bundles can be classified also by means of homotopy classes of analytic maps of X into classifying spaces.Among the others results, are worth recording: a relative approximation theorem of continuous cross sections by analytic ones, a theorem about the equivalence between analytical and topological homotopy between cross sections and a covering homotopy theorem.

Lie algebras of vector fields and generalized foliations

The main result is a Pursell-Shanks type theorem describing isomorphism of the Lie algebras of vector fields preserving generalized foliations. The result includes as well smooth as real-analytic and holomorphic cases.

Rigidity for Anosov Actions of Higher Rank Lattices

This is the simplest example of a large class of analytic actions of lattices in semisimple Lie groups on locally homogeneous spaces. A basic problem is to understand the differentiable actions near to such a standard action in terms of their geometry and dynamics (cf. [13], [50], [51]). A Cr-action So: IF > X -x X of a group F on a compact manifold X is said to be Anosov if there exists at least one element, Yh E F, such that 'p(Yh) is an Anosov diffeomorphism of X. We begin in this paper to study the Anosov differentiable actions of lattices, including many standard algebraic examples, and especially to study their properties. Our main theme is that either the Cr-rigidity or the Cr-deformation rigidity of an Anosov action (for 1 < r < cr. or even for the real analytic case) can be shown just by studying the behavior of the periodic orbits for the action. There are two notions of structural stability that appear in this paper, rigidity and deformation rigidity. A Cl-perturbation of a Cr-action So is simply another Cr-action Spl such that for a finite set of generators {81, . . , ad of F, the Cr-diffeomorphisms 'P(8i) and 'Pl(8i) are Cl-close for all i. An action So is said to be Cr-rigid (or topologically rigid if r = 0) if every sufficiently small C'perturbation of So is Cr-conjugate to So, for 0 < r < oo, or r = w in the case of real analytic actions. A Cl-deformation of an action So is a continuous path of Cr-actions (Pt defined for some 0 < t < e with Spo = SD* An action SD is said to be Cr-deform-

A spectral sequence for pseudogroups on 𝑅

Consider a pseudogroup P P of local homeomorphisms of R \mathbb {R} satisfying the following property: given points x 0 &gt; ⋯ &gt; x p {x_0} &gt; \cdots &gt; {x_p} and y 0 &gt; ⋯ &gt; y p {y_0} &gt; \cdots &gt; {y_p} in R \mathbb {R} , there is an element of P P , with domain an interval containing [ x 0 , x p ] [{x_0},{x_p}] , taking each x i {x_i} to y i {y_i} . The pseudogroup P r {P^r} of local C r {C^r} homeomorphisms, 0 ≤ r ≤ ∞ 0 \leq r \leq \infty , is of this type as is the pseudogroup P ω {P^\omega } of local real-analytic homeomorphisms. Let Γ \Gamma be the topological groupoid of germs of elements of P P . We construct a spectral sequence which involves the homology of a sequence of discrete groups { G p } \{ {G_p}\} . Consider the set { f ∈ P | f ( i ) = i , i = 0 , 1 , … , p } \{ f \in P|f(i) = i,i = 0,1, \ldots ,p\} ,; define f ∼ g f\sim g if f f and g g agree on a neighborhood of [ 0 , p ] ⊂ R [0,p] \subset \mathbb {R} . The equivalence classes under composition form the group G p {G_p} . Theorem: There is a spectral sequence with E p , q 1 = H q ( B G p ) E_{p,q}^1 = {H_q}(B{G_p}) which converges to H p + q ( B Γ ) {H_{p + q}}(B\Gamma ) . Our spectral sequence can be considered to be a version which covers the realanalytic case of some well-known theorems of J. Mather and G. Segal. The article includes some observations about how the spectral sequence applies to B Γ 1 ω B\Gamma _1^\omega . Further applications will appear separately.

From a differentiable to a real analytic perturbation theory, applications to the Kupka Smale theorems

AbstractKupka-Smale like theorems are proven in the real analytic case, using existing perturbation schemes for the smooth case and the heat operator. As a consequence, a topological proof is obtained of Siegel's theorem on the generic divergence of normal form transformations.

Formal properties of boundary-value problems for overdetermined systems of partial differential equations

For the pair (A,B), where A is a differential operator and B is a boundary operator in sections of bundles over the manifold M with boundary Γ, one indicates the construction of a complex of sheaves of germs of sections of the bundles Ei, over M and Gi over Γ and of differential-boundary operators $$\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{E} _o \xrightarrow{{(A, B)}} \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{E} _1 \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} _1 \to \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{E} _2 \times \underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{G} _2 \to \ldots \to 0 .$$ In the real analytic case the cohomology of this complex in the termE 1 ×G 1 is equal to zero (the local solvability of the boundary value problem under the compatibility conditions) if the conormal to the boundary Γ is in the direction of the least degeneracy of the symbol of the operator A, while the complex is exact if the degeneracy of the symbol is the same in all directions (for example, if A is elliptic).