Banach流形上映射度

本文讨论一种无限维流形上拓扑不变量，即具有可定向Ｆｒｅｄｈｏｌｍ结构的实Ｂａｎａｃｈ流形上　　Ｃｒ　　映射度．它是通常有限维流形上光滑映射度的一种自然推广．     

Differential calculus over general base fields and rings

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's C^k_c-theory.     

Mesonic Differential Forms

When M is a differentiable manifold, the exterior differential k -forms on M are the alternate k -linear forms on the tangent bundle T(M) . The mesonic differential k -forms are the k -linear forms on T(M) that are alternate with respect to the variables of odd rank, and also alternate with respect to the variables of even rank. After a reminder about meson algebras, and after the presentation of elementary properties of mesonic forms, this article introduces the mesonic differentiation of mesonic forms, which can be partially compared to the exterior differentiation of exterior forms. Some applications to riemannian manifolds and flat manifolds follow.     

Linearization of class C for contractions on Banach spaces

In this work we prove a C¹-linearization result for contraction diffeomorphisms, near a fixed point, valid in infinite-dimensional Banach spaces. As an intermediate step, we prove a specific result of existence of invariant manifolds, which can be interesting by itself and that was needed on the proof of our main theorem. Our results essentially generalize some classical results by P. Hartman in finite dimensions, and a result of Mora-Sola-Morales in the infinite-dimensional case. It is shown that the result can be applied to some abstract systems of semilinear damped wave equations.     

The stable manifold theorem for non-linear stochastic systems with memory: II. The local stable manifold theorem

We state and prove a Local Stable Manifold Theorem (Theorem 4.1) for non-linear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary trajectories of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary trajectory. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques developed by D. Ruelle, together with interpolation arguments.     

Chaotic weighted shifts in Bargmann space

This paper deals with the unilateral backward shift operator T on a Bargmann space F(C). This space can be identified with the sequence space ?²(N). We use the hypercyclicity criterion of Bès, Chan, and Seubert and the program of K.-G. Grosse-Erdmann to give a necessary and sufficient condition in order that T be a chaotic operator. The chaoticity of differentiation which correspond to the annihilation operator in quantum radiation field theory is in view, since the Bargmann space is an infinite-dimensional separable complex Hilbert space.     

An index theorem for Toeplitz operators on odd-dimensional manifolds with boundary

We establish an index theorem for Toeplitz operators on odd-dimensional spin manifolds with boundary. It may be thought of as an odd-dimensional analogue of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. In particular, there occurs naturally an invariant of η type associated to K¹ representatives on even-dimensional manifolds, which should be of independent interests. For example, it gives an intrinsic interpretation of the so called Wess-Zumino term in the WZW theory in physics.     

Vanishing theorems for ach Kähler manifolds and harmonic maps

We discuss a class of complete Kähler manifolds which are asymptotically complex hyperbolic near infinity. The main result is vanishing theorems for the second L ² cohomology of such manifolds when it has positive spectrum. We also generalize the result to the weighted Poincaré inequality case and establish a vanishing theorem provided that the weighted function ρ is of sub-quadratic growth of the distance function. We also obtain a vanishing theorem of harmonic maps on manifolds which satisfies the weighted Poincaré inequality.     

Long-time limit for a class of quadratic infinite-dimensional dynamical systems inspired by models of viscoelastic fluids

We study a class of quadratic, infinite-dimensional dynamical systems, inspired by models for viscoelastic fluids. We prove that these equations define a semi-flow on the cone of positive, essentially bounded functions. As time tends to infinity, the solutions tend to an equilibrium manifold in the L²-norm. Convergence to a particular function on the equilibrium manifold is only proved under additional assumptions. We discuss several possible generalizations.     

Normal systems over ANR’s, rigid embeddings and nonseparable absorbing sets

Most of results of Bestvina and Mogilski [Characterizing certain incomplete infinite-di-mensional absolute retracts. Michigan Math. J., 33, 291-313 (1986)] on strong Z-sets in ANR's and absorbing sets is generalized to nonseparable case. It is shown that if an ANR X is locally homotopy dense embeddable in infinite-dimensional Hilbert manifolds and w(U ) = w(X ) (where "w"is the topological weight) for each open nonempty subset U of X , then X itself is homotopy dense embeddable in a Hilbert manifold. It is also demonstrated that whenever X is an AR, its weak product W (X, ) = {(x n ) ∞ n=1 ∈ X ω : x n = for almost all n} is homeomorphic to a pre-Hilbert space E with E ～ = ΣE. An intrinsic characterization of manifolds modelled on such pre-Hilbert spaces is given.     

A generalization of Aubry–Mather theory to partial differential equations and pseudo-differential equations

We discuss an Aubry–Mather-type theory for solutions of non-linear, possibly degenerate, elliptic PDEs and other pseudo-differential operators.We show that for certain PDEs and ΨDEs with periodic coefficients and a variational structure it is possible to find quasi-periodic solutions for all frequencies. This results also hold under a generalized definition of periodicity that makes it possible to consider problems in covers of several manifolds, including manifolds with non-commutative fundamental groups.An abstract result will be provided, from which an Aubry–Mather-type theory for concrete models will be derived.     

Regularity ofp-energy minimizing maps andp-superstrongly unstable indices

We derive the first and the second variational formulas forp-energy functional on maps between Riemannian manifolds, obtain a Bochner formula with related estimates and discuss Liouville-type theorems and the regularity ofp-minimizers. In particular, via an extrinsic average variational method,p-superstrongly unstable manifolds and indices are found and their role in the regularity theory is established.     

Unitary representations of unimodular Lie groups in Bergman spaces

For an arbitrary unimodular Lie group G, we construct strongly continuous unitary representations in the Bergman space of a strongly pseudoconvex neighborhood of G in the complexification of its underlying manifold. These representation spaces are infinite-dimensional and have compact kernels. In particular, the Bergman spaces of these natural manifolds are infinite-dimensional.     

Hamiltonian Gromov-Witten invariants

In this paper we introduce invariants of semi-free Hamiltonian actions of S¹ on compact symplectic manifolds using the space of solutions to certain gauge theoretical equations. These equations generalise both the vortex equations and the holomorphicity equation used in Gromov-Witten theory. In the definition of the invariants we combine ideas coming from gauge theory and the ideas underlying the construction of Gromov-Witten invariants.     

Strichartz estimates and the nonlinear Schr?dinger equation on manifolds with boundary

We establish Strichartz estimates for the Schr?dinger equation on Riemannian manifolds (Ω, g) with boundary, for both the compact case and the case that Ω is the exterior of a smooth, non-trapping obstacle in Euclidean space. The estimates for exterior domains are scale invariant; the range of Lebesgue exponents (p, q) for which we obtain these estimates is smaller than the range known for Euclidean space, but includes the key ${L^{4}_{t}L^{\infty}_x}$ estimate, which we use to give a simple proof of well-posedness results for the energy critical Schr?dinger equation in 3 dimensions. Our estimates on compact manifolds involve a loss of derivatives with respect to the scale invariant index. We use these to establish well-posedness for finite energy data of certain semilinear Schr?dinger equations on general compact manifolds with boundary.     

On conformally flat Lorentz parabolic manifolds

We introduce conformally flat Fefferman-Lorentz manifold of parabolic type as a special class of Lorentz parabolic manifolds. It is a smooth (2n+2)-manifold locally modeled on (Û(n+1, 1), S ^2n+1,1). As the terminology suggests, when a Fefferman-Lorentz manifold M is conformally flat, M is a Fefferman-Lorentz manifold of parabolic type. We shall discuss which compact manifolds occur as a conformally flat Fefferman-Lorentz manifold of parabolic type.     

On locally conformal almost Kähler manifolds

In the first section of this note, we discuss locally conformal symplectic manifolds, which are differentiable manifoldsV ²ⁿ endowed with a nondegenerate 2-form Ω such thatdΩ=θ ∧ Ω for some closed form θ. Examples and several geometric properties are obtained, especially for the case whendΩ ≠ 0 at every point. In the second section, we discuss the case when Ω above is the fundamental form of an (almost) Hermitian manifold, i.e. the case of the locally conformal (almost) Kähler manifolds. Characterizations of such manifolds are given. Particularly, the locally conformal Kähler manifolds are almost Hermitian manifolds for which some canonically associated connection (called the Weyl connection) is almost complex. Examples of locally conformal (almost) Kähler manifolds which are not globally conformal (almost) Kähler are given. One such example is provided by the well-known Hopf manifolds.     

The Witten deformation for even dimensional spaces with cone-like singularities and admissible Morse functions

In this Note we generalise the Witten deformation to even dimensional Riemannian manifolds with cone-like singularities X and certain functions f, which we call admissible Morse functions. As a corollary we get Morse inequalities for the L²-Betti numbers of X. The contribution of a singular point p of X to the Morse inequalities can be expressed in terms of the intersection cohomology of the local Morse datum of f at p. The definition of the class of functions which we study here is inspired by stratified Morse theory as developed by Goresky and MacPherson. However the setting here is different since the spaces considered here are manifolds with cone-like singularities instead of Whitney stratified spaces.     

Stable Manifolds and Nonuniform Hyperbolicity: Optimal Estimates

We establish the existence of invariant stable manifolds for C ¹ perturbations of a nonuniform exponential dichotomy with an arbitrary nonuniform part. We consider the general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. We also obtain optimal estimates for the decay of trajectories along the stable manifolds. The optimal C ¹ smoothness of the invariant manifolds is obtained using an invariant family of cones.     

Gromov-Hausdorff convergence to nonmanifolds

We construct sequences of S⁷’s in Gromov-Hausdorff space converging to nonmanifolds. The manifolds in these sequences have a common contractibility function. There are two main classes of examples—examples for which the limit is an infinite-dimensional space of finite cohomological dimension and examples for which the limit is a finite-dimensional ANR homology manifold. Communicated by Karsten Grove