Statistical Estimation of Generalized Dimensions

A statistical estimate for generalized dimensions of a set based on the computation of average distances to the closest points in a sample of elements of A is given. For smooth manifolds with Lebesgue measures and for self-similar fractals with self-similar measures, the estimate is proved to be consistent.     

Complexifications and real forms of Hamiltonian structures

We consider generalizations of the standard Hamiltonian dynamics to complex dynamical variables and introduce the notions of real Hamiltonian form in analogy with the notion of real forms for a simple Lie algebra. Thus to each real Hamiltonian system we are able to relate several nonequivalent ones. On the example of the complex Toda chain we demonstrate how starting from a known integrable Hamiltonian system (e.g. the Toda chain) one can complexify it and then project onto different real forms. Received 18 October 2001 / Received in final form 24 May 2002 Published online 2 October 2002 RID="a" ID="a"e-mail: gerjikov@inrne.bas.bg     

Fréchet Algebra Techniques for Boundary Value Problems on Noncompact Manifolds: Fredholm Criteria and Functional Calculus via Spectral Invariance

A Boutet de Monvel type calculus is developed for boundary value problems on (possibly) noncompact manifolds. It is based on a class of weighted symbols and Sobolev spaces. If the underlying manifold is compact, one recovers the standard calculus. The following is proven:

• 1 The algebra G of Green operators of order and type zero is a spectrally invariant Fréchet subalgebra of L(H), H a suitable Hilbert space, i. e.,
• 2 Focusing on the elements of order and type zero is no restriction since there are order reducing operators within the calculus.
• 3 There is a necessary and sufficient criterion for the Fredholm property of boundary value problems, based on the invertibility of symbols modulo lower order symbols, and
• 4 There is a holomorphic functional calculus for the elements of G in several complex variables.

     

Compact complex threefolds which are Kähler outside a smooth rational curve

Let M be a compact complex manifold containing an irreducible curve C such that M — C is Kähler; in this paper we study the link between some cohomological properties of C and the obstructions to the existence of a Kähler metric on the whole of M. In particular we get that, if M is not Kähler, then C is a $ \left({\partial + \bar \partial} \right) $ –exact current, or there exists a positive current S of bidimension (1, 1) such that $ \partial \bar \partial S = 0,\,\chi _C S = 0 $ and S + C is $ \left({\partial + \bar \partial} \right) $ –exact. If C is a smooth rational curve, more precise results are given in connection with the normal bundle N_C|M.     

Lipschitz precompactness for closed negatively curved manifolds

We prove that, given a integer and a group , the class of closed Riemannian -manifolds of uniformly bounded negative sectional curvatures and with fundamental groups isomorphic to is precompact in the Lipschitz topology. In particular, the class breaks into finitely many diffeomorphism types.

     

The groups of quasiconformal homeomorphisms on Riemann surfaces

Suppose is a connected Riemann surface. Let denote the homeomorphism group of with the compact-open topology, and denote the subgroup of quasiconformal mappings of onto itself, and let and denote the identity components of and respectively. In this paper we show that the pair is an -manifold, and determine their topological types.

     

Invariant Bidifferential Operators on Tensor Densities over a Contact Manifold

The spaces of tensor densities over a manifold M are modules over the Lie algebra Vect(M) of vector fields over the manifold. When M is a contact manifold, one can consider the algebra C(M) of vector fields which preserves the contact structure. If the manifold is endowed with a contact projective structure, there is an embedding of the linear symplectic algebra sp(2n+2, ) in C(M). In this Letter, we determine the C(M)- and the sp(2n+2, )-invariant bidifferential operators on tensor densities.     

Renormalization and Dimensional Regularization for a Scalar Field with Gauss–Bonnet-Type Coupling to Curvature

We consider a scalar field with a Gauss–Bonnet-type coupling to the curvature in a curved space–time. For such a quadratic coupling to the curvature, the metric energy–momentum tensor does not contain derivatives of the metric of orders greater than two. We obtain the metric energy–momentum tensor and find the geometric structure of the first three counterterms to the vacuum averages of the energy–momentum tensors for an arbitrary background metric of an N-dimensional space–time. In a homogeneous isotropic space, we obtain the first three counterterms of the n-wave procedure, which allow calculating the renormalized values of the vacuum averages of the energy–momentum tensors in the dimensions N=4,5. Using dimensional regularization, we establish that the geometric structures of the counterterms in the n-wave procedure coincide with those in the effective action method.     

Two-weight Caccioppoli inequalities for solutions of nonhomogeneous -harmonic equations on Riemannian manifolds

In this paper, we first prove the local two-weight Caccioppoli inequalities for solutions to the nonhomogeneous -harmonic equation of the form . Then, as applications of the local results, we prove the global two-weight Caccioppoli-type inequalities for these solutions on Riemannian manifolds.