Banach Manifolds Research Articles

Unbounded Fredholm Modules and Spectral Flow

AbstractAn odd unbounded (respectively, p-summable) Fredholm module for a unital Banach *-algebra, A, is a pair (H,D) where A is represented on the Hilbert space, H, and D is an unbounded self-adjoint operator on H satisfying:(1) (1 + D2)-1 is compact (respectively, Trace_(1 + D2)-(p/2)_∞), and(2) ﹛a ∈ A | [D, a] is bounded﹜ is a dense *- subalgebra of A.If u is a unitary in the dense *-subalgebra mentioned in (2) thenuDu* = D + u[D, u*] = D + Bwhere B is a bounded self-adjoint operator. The pathis a “continuous” path of unbounded self-adjoint “Fredholm” operators. More precisely, we show thatis a norm-continuous path of (bounded) self-adjoint Fredholm operators. The spectral flow of this path is roughly speaking the net number of eigenvalues that pass through 0 in the positive direction as t runs from 0 to 1. This integer,recovers the pairing of the K-homology class [D] with the K-theory class [u].We use I.M. Singer's idea (as did E. Getzler in the θ-summable case) to consider the operator B as a parameter in the Banach manifold, Bsa(H), so that spectral flow can be exhibited as the integral of a closed 1-formon this manifold. Now, for B in ourmanifold, any X ∈ TB_Bsa(H)_ is given by an X in Bsa(H) as the derivative at B along the curve t→ B + tX in the manifold. Then we show that for m a sufficiently large half-integer:is a closed 1-form. For any piecewise smooth path {Dt = D + Bt} with D0 and D1 unitarily equivalent we show thatthe integral of the 1-form ã. If D0 and D1 are not unitarily equivalent, wemust add a pair of correction terms to the right-hand side. We also prove a bounded finitely summable version of the form:for an integer. The unbounded case is proved by reducing to the bounded case via the map . We prove simultaneously a type II version of our results.

The hausdorff dimension of the harmonic class on negatively curved surfaces

We study the regularity of the Hausdorff dimension of the harmonic class of a surface M of negative curvature as a function of the riemannian metric. We prove that it is a Cr− 3 function of the metric in the Banach manifold of Cr riemannian metrics on M. We also prove regularity results for some asymptotic quantities associated to the Brownian motion on M.

On the disc-convexity of complex Banach manifolds

The Banach hyperbolicity and disc-convexity of complex Banach manifolds and their relations are investigated.

Classical Solutions of Multidimensional Hele--Shaw Models

Existence and uniqueness of classical solutions for the multidimensional expanding Hele{Shaw problem are proved.

Ratio geometry, rigidity and the scenery process for hyperbolic Cantor sets

Given a ${\cal C}^{1+\gamma}$ hyperbolic Cantor set $C$, we study the sequence $C_{n,x}$ of Cantor subsets which nest down toward a point $x$ in $C$. We show that $C_{n,x}$ is asymptotically equal to an ergodic Cantor set valued process. The values of this process, called limit sets, are indexed by a Hölder continuous set-valued function defined on Sullivan's dual Cantor set. We show the limit sets are themselves ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets, with the highest degree of smoothness which occurs in the ${\cal C}^{1+\gamma}$ conjugacy class of $C$. The proof of this leads to the following rigidity theorem: if two ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$ hyperbolic Cantor sets are ${\cal C}^1$ conjugate, then the conjugacy (with a different extension) is in fact already ${\cal C}^{k+\gamma},{\cal C}^\infty$ or ${\cal C}^\omega$. Within one ${\cal C}^{1+\gamma}$ conjugacy class, each smoothness class is a Banach manifold, which is acted on by the semigroup given by rescaling subintervals. Smoothness classes nest down, and contained in the intersection of them all is a compact set which is the attractor for the semigroup: the collection of limit sets. Convergence is exponentially fast, in the ${\cal C}^1$ norm.

Electrical network theory of countable graphs

The purpose of this dissertation is to derive the equations governing electrical networks of countable graphs and to give conditions assuring that these are gradient dynamical systems on semi-Riemannian Hilbert manifolds. Furthermore, we show how symmetries in the graph give rise to a G-action on the manifold of states. Finally, we present some mathematical results about gradient dynamical systems on semi-Riemannian Banach manifolds.

Minimax theorems and its applications in locally convex subsets of banach manifolds

Minimax theorems and its applications in locally convex subsets of banach manifolds

CkSmoothness of Invariant Curves in a Global Saddle-Node Bifurcation

The birth ofCk-smooth invariant curves from a saddle-node bifurcation in a family ofCkdiffeomorphisms on a Banach manifold (possibly infinite dimensional) is constructed in the case that the fixed point is a stable node along hyperbolic directions, and has a smooth noncritical curve of homoclinic orbits. This ensures that the map restricted to the resulting curve is equivalent to aCkmap of the circle. In particular, for aC2family of diffeomorphisms the resulting curve isC2, and the “Denjoy example” cannot occur. Included is a new smoothness result for the foliation transversal to the center subspace, for the finite and infinite dimensional cases. Specifically,Ck-smoothness with respect to all variables of invariant foliations of the center-stable and center-unstable manifolds of a partially hyperbolic fixed point is proved in all cases.

Holomorphic maps of uniform type

(E,X)to hold. This problem for vector-valued holomorphic maps, i.e. for the casewhere X is a locally convex space, was investigated by some authors. Thefirst result on this problem belongs to Colombeau and Mujica. In [2] theyhave shown that the equality (UN) holds when E is a dual Fr´echet–Montelspace and X a Fr´echet space. Next, a necessary and sufficient conditionfor (UN) to hold in the class of scalar holomorphic functions on a nuclearFr´echet space was established by Meise and Vogt [7]. An important sufficientcondition for (UN) for scalar holomorphic functions on such a space was alsofound recently by those two authors [8]. However, until now, when X doesnot have a linear structure, the problem has not been investigated.Here we consider this problem for holomorphic maps with values in acomplex manifold of infinite dimension, in particular, in the projective spaceassociated with a Fr´echet space (see the definition in §2). In the first section,by the method of [4], we give a characterization of the uniformity of holo-morphic maps with values in complex Banach manifolds. The scalar casehas been proved by Meise and Vogt [7] by a different method. Section 2is devoted to proving the main result (Theorem 2.1) of this note: everyholomorphic map from a dual space of a nuclear Fr´echet space (i.e., from

Weighted l1 ‐ Grassmannians and Banach Manifolds of Solutions of the KP‐ Equation and the KdV‐Equation

Weighted <i>l</i>₁ ‐ Grassmannians and Banach Manifolds of Solutions of the KP‐ Equation and the KdV‐Equation

A criterion for Banach manifolds to be finite-dimensional

We extend the results obtained in [1] to the case of arbitrary Banach spaces and manifolds. We give an example of a continuous bijective mapping with discontinuous inverse which acts in a Banach space and differs from the identical mapping only in an open unit ball. A criterion for a Banach manifold to be finite-dimensional is established in terms of the continuity of inverse operators.

Right inverses of vector fields

AbstractD. Przeworska-Rolewicz developed an algebra-based theory around linear, not necessarily continuous, operators D: X → X which admit a right inverse, the elementary example being D = d/dt or, more generally, where ai are constants. We give conditions for the right invertibility of D in the case where ai are functions, or more generally, where D is the Lie or convariant derivative associated with a vector field on a (Banach) manifold M.

Partially hyperbolic sets from a co-dimension one bifurcation

We study the saddle-node bifurcation of a partially hyperbolic fixed point in a Lipschitz family of $C^{k}$ diffeomorphisms on a Banach manifold (possibly infinite dimensional) in the case that the fixed point is a saddle along hyperbolic directions and has multiple curves of homoclinic orbits. We show that this bifurcation results in an invariant set which consists of a countable collection of closed invariant curves and an uncountable collection of nonclosed invariant curves which are the topological limits of the closed curves. In addition, it is shown that these curves are $C^k$-smooth and that this invariant set is uniformly partially hyperbolic.

Orientability of Fredholm Families and Topological Degree for Orientable Nonlinear Fredholm Mappings

We construct a degree theory for oriented Fredholm mappings of index zero between open subsets of Banach spaces and between Banach manifolds. Our approach is based on the orientation of Fredholm mappings: it does not use Fredholm structures on the domain and target spaces. We provide a computable formula for the change in degree through an admissible homotopy that is necessary for applications to global bifurcation. The notion of orientation enables us to establish rather precise relationships between our degree and many other degree theories for particular classes of Fredholm maps, including the Elworthy-Tromba degree, which have appeared in the literature in a seemingly unrelated manner.

A criterion for Banach manifolds with a separable model to be finite-dimensional

We give an example of a continuous bijective mapping with a discontinuous inverse which acts in a separable Banach space and differs from the identical mapping only in an open unit ball. A criterion for Banach manifolds with a separable model to be finite-dimensional is established in terms of the continuity of inverse operators.

The Kobayashi and Carathéodory pseudodistances for complex analytic manifolds

Pseudodistances defined on analytic Banach manifolds permit one to obtain a number of results on that space by purely topological methods. In addition, they enable one to give geometric insight into function theoretic results. In this paper, we extend the notion of a complex analytic manifold to a complex analytic Banach manifold over a complex Banach space. Then we show that if M is a complex analytic Banach manifold, then the Kobayashi pseudodistance is the largest for which every holomorphic mapping from the unit disk of the complex plane into a complex analytic Banach manifold is distance decreasing, whereas the Carathéodory pseudodistance is the smallest pseudodistance from which every holomorphic mapping from the complex analytic manifold to the unit disk of the complex plane is distance decreasing.

The manifold structure of maps between open manifolds

We establish in a canonical manner a manifold structure for the completed space of bounded maps between open manifoldsM andN, assuming thatM andN are endowed with Riemannian metrics of bounded geometry up to a certain order. The identity component of the corresponding diffeomorphisms is a Banach manifold and metrizable topological group.

The space of spectral measures is a homogeneous reductive space

We prove that the space of spectral measures on a W*-algebra is a smooth Banach manifold in a natural way and that the action of the group of invertible elements of the algebra by inner automorphisms makes it into a reductive homogeneous space. This gives a geometric structure for the set of normal operators with the same spectrum.

A Min-max Principle with a Relaxed Boundary Condition

A standard Min-Max procedure to find critical points for a ${C^1}$functional $\varphi$ verifying a compactness condition of Palais-Smale type on a smooth Banach manifold $X$ consists of finding an appropriate class $\mathcal {F}$ of compact subsets of $X$, all containing a fixed boundary $B$, and then showing that the value $c = {\inf _{A \in \mathcal {F}}}{\sup _{x \in A}}\varphi (x)$ is a critical level, provided it satisfies $\sup \varphi (B) < c$. In this paper, we refine this procedure by relaxing the boundary condition.

Holomorphic families of geodesic discs in infinite dimensional Teichmüller spaces

The theory of quasiconformal mappings plays an important role in Teichmüller theory. The Teichmüller spaces of Riemann surfaces are defined as quotient spaces of the spaces of Beltrami differentials, and the Teichmüller distances are defined to measure quasiconformal deformations between the Riemann surfaces representing points in the Teichmüller spaces. The Teichmüller spaces are complex Banach manifolds equipped with natural complex structures such that the canonical projections are holomorphic. It is known (see Gardinar [4]) that the Teichmüller distance, defined independently of the complex structures, coincides with the Kobayashi distance.In spite of the naturality of the definition of a Teichmüller space as a quotient of Beltrami differentials, for given two Beltrami differentials it is hard to determine whether they are equivalent or not. For this reason, it is not trivial to describe geodesic lines with respect to the Teichmüller-Kobayashi metric.