Foliations of asymptotically flat manifolds by surfaces of Willmore type

The goal of this paper is to establish the existence of a foliation of the asymptotic region of an asymptotically flat manifold with positive mass by surfaces which are critical points of the Willmore functional subject to an area constraint. Equivalently these surfaces are critical points of the Geroch–Hawking mass. Thus our result has applications in the theory of general relativity.     

Foliations transverse to fibers of Seifert manifolds

In this paper we prove the conjecture of Jankins and Neumann [JN2] about rotation numbers of products of circle homeomorphisms, which together with other results of [EHN] and [JN2] (mentioned below) implies that a Seifert manifold admits foliations tranverse to its fibers only if it admits such foliations with a projective transverse structure. 1991Mathematics Subject Classification: Primary 55R05, 57R30, Secondary 47A35, 57M99, 58F11. Partially supported by a Sloan Doctoral Dissertation Fellowship, and by the Technion-Israel Institute of Technology.     

Foliations with a Kupka component on algebraic manifolds

We consider codimension one holomorphic foliations in complex projective manifolds of dimension at least 3, having a compact Kupka component and represented by integrable holomorphic sections of the bundleTM ^*L, whereL denotes a very ample holomorphic line bundle. We will show that, if the transversal type is not the radial vector field andH ¹ (M,) = 0, then the foliation has a meromorphic first integral.Supported by Conacyt: 3398-E 9307     

Foliations on the tangent bundle of Finsler manifolds

Let M be a smooth manifold with Finsler metric F,and let T M be the slit tangent bundle of M with a generalized Riemannian metric G,which is induced by F.In this paper,we prove that (i) (M,F) is a Landsberg manifold if and only if the vertical foliation F V is totally geodesic in (T M,G);(ii) letting a:= a(τ) be a positive function of τ=F 2 and k,c be two positive numbers such that c=2 k(1+a),then (M,F) is of constant curvature k if and only if the restriction of G on the c-indicatrix bundle IM (c) is bundle-like for the horizontal Liouville foliation on IM (c),if and only if the horizontal Liouville vector field is a Killing vector field on (IM (c),G),if and only if the curvature-angular form Λ of (M,F) satisfies Λ=1-a 2/R on IM (c).     

On manifolds swept out by high dimensional quadrics

We give a completely different, much shorter, proof of a substantial generalization of the main result from Kachi and Sato (Mem. Am. Math. Soc. 160:763, 2002). It states that embedded projective n-folds swept out by quadrics of dimension at least are either scrolls or hyperquadric fibrations, which are also Mori contractions. An erratum to this article can be found at     

Quadratic forms of closed manifolds

In this paper it is shown that for any integral-valued unimodular quadratic form and any number n of the form 8k + 4 (where k1), there exists a smooth closed n-dimensional manifold with this quadratic form. The proof is based on the construction (with the help of the plumbing construction) of smooth closed three-connected eight-dimensional manifolds with given form.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 122, pp. 104–108, 1982.     

Brownian motion in infinite dimensional manifolds

We discuss the extension to infinite dimensional Riemannian—Wiener manifolds of the transport approximation to Brownian motion, which was formulated by M. Pinsky for finite dimensional manifolds. A global representation is given for the Laplace—Beltrami operator in terms of the Riemannian spray and a homogenizing operator based upon the central hitting measure of the surface of the unit ball with respect to the Brownian motion on the model space.Research supported by NSF grant MCS8202319.     

Three dimensional homogeneous Finsler manifolds

In this paper we study three dimensional homogeneous Finsler manifolds. We first obtain a complete list of the three‐dimensional homogeneous manifolds which admit invariant Finsler metrics. Then we consider invariant Randers metrics and present the classification of three dimensional homogeneous Randers spaces under isometrics.     

Surgery on pairs of closed manifolds

To apply surgery theory to the problem of classifying pairs of closed manifolds, it is necessary to know the subgroup of the group LP* generated by those elements which are realized by normal maps to a pair of closed manifolds. This closely relates to the surgery problem for a closed manifold and to the computation of the assembly map. In this paper we completely determine such subgroups for many cases of Browder-Livesay pairs of closed manifolds. Moreover, very explicit results are obtained in the case of an elementary fundamental group. Then we generalize them, and obtain several further results about the realization of elements in the Browder-Quinn surgery obstruction groups by means of normal maps to a closed manifold filtered by closed submanifolds. Partially supported by the RFFI Grant No. 05-01-00993, by the GNSAGA of the National Research Council of Italy, by the MIUR (Ministero della Istruzione, Università e Ricerca) of Italy within the project ‘Proprietà Geometriche delle Varietà Reali e Complesse’, and by a Research Grant of the University of Modena and Reggio Emilia. The second author would like to thank the MPI (Bonn), the ICTP (Trieste), and the University of Modena and Reggio Emilia for their kind hospitality and support in different periods during the year 2007, and the Commission on Development and Exchange of IMU for the travel support.     

Self-duality operators on odd dimensional manifolds

In this paper we construct a new elliptic operator associated to any nowhere zero vector field on an odd-dimensional manifold and study its index theory. It turns out this operator has several geometric applications to conformal vector fields, self-dual vector fields, locally free -actions and transversal hypersurfaces of these vector fields in an odd-dimensional manifold. In particular, we reveal a non-stable phenomena about the existence of conformal vector fields and self-dual vector fields in odd dimensions above 3. This is in sharp contrast to the stable phenomena about the existence of nowhere zero vector fields in odd dimensions. Besides these applications, the index formula of this new operator also gives the formulas for the dimensions of self-duality cohomology groups and for the virtual dimensions of the moduli spaces of anti-self-dual connections on 5-cobordisms, which are introduced in author's previous papers.

     

Measures on infinite dimensional Grassmann manifolds

A 1-parameter family of quasi-invariant measures is presented. These measures are cylinder measures in graph coordinates. Their characteristic functions are represented as integrals relative to an infinite product measure. This is applied to the problem of determining the support properties of the measures. One of the measures can be used to define the unitary structure for the basic representation of the affine extension of the restricted unitary group.     

Iteration of closed geodesics in stationary Lorentzian manifolds

Following the lines of Bott in (Commun Pure Appl Math 9:171–206, 1956), we study the Morse index of the iterates of a closed geodesic in stationary Lorentzian manifolds, or, more generally, of a closed Lorentzian geodesic that admits a timelike periodic Jacobi field. Given one such closed geodesic γ, we prove the existence of a locally constant integer valued map Λ_γ on the unit circle with the property that the Morse index of the iterated γ ^N is equal, up to a correction term ε_γ∈{0,1}, to the sum of the values of Λ_γ at the N-th roots of unity. The discontinuities of Λ_γ occur at a finite number of points of the unit circle, that are special eigenvalues of the linearized Poincaré map of γ. We discuss some applications of the theory.     

Foliations induced by congruences

Communicated by Klaus Keimel