Center manifolds of infinite dimensions II: Proofs of the main results

In this paper, the part II of a bipartite work, we present the proof of Theorem 1 which was stated in part I, i.e. the precursor of the present paper. This proof establishes the existence of a smooth center manifold, i.e. of a smooth decoupling function for the abstract evolution system considered in part I. In an appendix, the proofs of some auxiliary lemmas are presented, some of which were stated in part I, while the others are related to the present proof of Theorem 1.     

Center manifolds for infinite delay

We obtain a C¹ center manifold theorem for perturbations of delay difference equations in Banach spaces with infinite delay. Our results extend in several directions the existing center manifold theorems. Besides considering infinite delay equations, we consider perturbations of nonuniform exponential trichotomies and generalized trichotomies that may exhibit stable, unstable and central behaviors with respect to arbitrary asymptotic rates ecρ(n) for some diverging sequence ρ(n). This includes as a very special case the usual exponential behavior with ρ(n)=n.     

The Dolbeault complex in infinite dimensions I

In this paper we introduce certain basic notions concerning infinite dimensional complex manifolds, and prove that the Dolbeault cohomology groups of infinite dimensional projective spaces, with values in finite rank vector bundles, vanish. Some applications of such vanishing theorems are discussed; e.g., we classify vector bundles of finite rank over infinite dimensional projective spaces. Finally, we prove a sharp theorem on solving the inhomogeneous Cauchy-Riemann equations on affine spaces.

     

Infinite horizon BSDEs in infinite dimensions with continuous driver and applications

This paper is devoted to forward-backward systems of stochastic differential equations in which the forward equation is not coupled to the backward one, both equations are infinite dimensional and on the time interval [0, + ∞). The forward equation defines an Ornstein-Uhlenbeck process, the driver of the backward equation has a linear part which is the generator of a strongly continuous, dissipative, compact semigroup, and a nonlinear part which is assumed to be continuous with linear growth. Under the assumption of equivalence of the laws of the solution to the forward equation, we prove the existence of a solution to the backward equation. We apply our results to a stochastic game problem with infinitely many players.     

Center manifolds and optimal estimates

For sufficiently small perturbations of a nonuniform exponential trichotomy, we establish the existence of $C^k$ invariant center manifolds. We consider the general case of sequences of maps, which corresponds to a nonautonomous dynamics with discrete time. In particular, we obtain optimal estimates for the decay of all derivatives along the trajectories on the center manifolds.     

Center manifolds theorem for parameterized delay differential equations with applications to zero singularities

The goal of this paper is to develop a center manifold theory for delay differential equations with parameters. As applications, we use the center manifold theorem to establish fold and Bogdanov-Takens bifurcations. In particular, we obtain the versal unfoldings of delayed predator-prey systems with predator harvesting at the Bogdanov-Takens singularity.     

Structure groups and holonomy in infinite dimensions

We give a theorem of reduction of the structure group of a principal bundle P with regular structure group G. Then, when G is in the classes of regular Lie groups defined by T. Robart in [Can. J. Math. 49 (4) (1997) 820-839], we define the closed holonomy group of a connection as the minimal closed Lie subgroup of G for which the previous theorem of reduction can be applied. We also prove an infinite dimensional version of the Ambrose-Singer theorem: the Lie algebra of the holonomy group is spanned by the curvature elements.     

Coverings and dimensions in infinite profinite groups

Answering a question of Miklós Abért, we prove that an infinite profinite group cannot be the union of less than continuum many translates of a compact subset of box dimension less than 1. Furthermore, we show that it is consistent with the axioms of set theory that in any infinite profinite group there exists a compact subset of Hausdorff dimension 0 such that one can cover the group by less than continuum many translates of it.     

Examples of Einstein manifolds in odd dimensions

We construct Einstein metrics of non-positive scalar curvature on certain solid torus bundles over a Fano Kähler–Einstein manifold. We show, among other things, that the negative Einstein metrics are conformally compact, and the Ricci-flat metrics have slower-than-Euclidean volume growth and quadratic curvature decay. Also we construct positive Einstein metrics on certain 3-sphere bundles over a Fano Kähler–Einstein manifold. We classify the homeomorphism and diffeomorphism types of the total spaces when the base is the complex projective plane.     

An infinite family of manifolds with bounded total curvature

The negative answer to the following problem of V. I. Arnold is given: Is the number of topologically different -manifolds of bounded total curvature finite?

     

The convergence-extension theorem of Noguchi in infinite dimensions

In this note we give generalizations of Noguchi's convergence-extension theorem to the case of infinite dimension.

     

Regularity properties of transition probabilities in infinite dimensions

In a Hilbert space H we consider a process X solution of a semilinear stochastic differential equation, driven by a Wiener process. We prove that, under appropriate conditions, the transition probabilities of X are absolutely continuous with respect to a properly chosen gaussian measure μ in H, and the corresponding densities belong to some Wiener-Sobolev spaces over (H,μ). In the linear caseX is a nonsymmetric Ornstein-Uhlenbeck process, with possibly degenerate diffusion coefficient. The general case is treated by the Girsanov. Theorem and the Malliavin calculus. Examples and applications to stochastic partial differential equations are given     

Center manifolds of dynamical systems under discretization

We consider an autonomous dynamical system discretized by a one-step method. The point z = 0 is assumed to be fixed under the continuous and the discrete flows. We allow z = 0 to be non-hyperbolic. The continuous system has a center-unstable manifold and we show the existence of approximating invariant manifolds for the discretizations. The manifolds for the continuous and the discrete systems share the property of being locally attracting at an exponential rate; the dynamics inside the manifolds can differ qualitatively, however, for all step-sizes h.