Negative exterior curvature foliations on compact Riemannian manifolds

The topological structure of compact Riemannian manifolds that admit hyperbolic foliations is studied. Translated fromMatematicheskie Zametki, Vol. 62, No. 5, pp. 673–676, November, 1997. Translated by S. S. Anisov     

Additivity for the parametrized topological Euler characteristic and Reidemeister torsion

Dwyer, Weiss, and Williams have recently defined the notions of the parametrized topological Euler characteristic and the parametrized topological Reidemeister torsion which are invariants of bundles of compact topological manifolds. We show that these invariants satisfy additivity formulas paralleling the additive properties of the classical Euler characteristic and the Reidemeister torsion of CW-complexes.     

Geometrically finite groups,Patterson-Sullivan measures and Ratner's ridigity theorem

Summary The rigidity properties of the horospherical foliations of geometrically finite hyperbolic manifolds are investigated. Ratner's theorem generalizes to these foliations with respect to the Patterson-Sullivan measure. In the spirit of Mostow, we prove the nonexistence of invariant measurable distributions on the boundary of hyperbolic space for geometrically finite groups. Finally, we show that the frame flow on geometrically finite hyperbolic manifolds is Bernoulli.partially supported by NSF Grant No. DMS-820-04024partially supported by NSF Grant No. DMS-85-02319     

Invariants of four- and three-dimensional singularities of integrable systems

A relationship between invariants of four-dimensional singularities of integrable Hamiltonian systems (with two degrees of freedom) and invariants of two-dimensional foliations on three-dimensional manifolds being the “boundaries” of these four-dimensional singularities is discovered. Nonequivalent singularities which, nevertheless, have equal three-dimensional invariants are found.     

Non-realizability and ending laminations: Proof of the density conjecture

We give a complete proof of the Bers?CSullivan?CThurston density conjecture. In the light of the ending lamination theorem, it suffices to prove that any collection of possible ending invariants is realized by some algebraic limit of geometrically finite hyperbolic manifolds. The ending invariants are either Riemann surfaces or filling laminations supporting Masur domain measured laminations and satisfying some mild additional conditions. With any set of ending invariants we associate a sequence of geometrically finite hyperbolic manifolds and prove that this sequence has a convergent subsequence. We derive the necessary compactness theorem combining the Rips machine with non-existence results for certain small actions on real trees of free products of surface groups and free groups. We prove then that the obtained algebraic limit has the desired conformal boundaries and the property that none of the filling laminations is realized by a pleated surface. In order to be able to apply the ending lamination theorem, we have to prove that this algebraic limit has the desired topological type and that these non-realized laminations are ending laminations. That this is the case is the main novel technical result of this paper. Loosely speaking, we show that a filling Masur domain lamination which is not realized is an ending lamination.     

Holomorphic Maps of Compact Generalized Hopf Manifolds

We investigate holomorphic maps between compact generalized Hopf manifolds (i.e., locally conformal Kähler manifolds with parallel Lee form). We show that they preserve the canonical foliations. Moreover, we study compact complex submanifolds of g.H. manifolds and holomorphic submersions from compact g.H. manifolds.     

Minimal volume invariants,topological sphere theorems and biorthogonal curvature on 4‐manifolds

The goal of this article is to establish estimates involving the Yamabe minimal volume, mixed minimal volume and some topological invariants on compact 4‐manifolds. In addition, we provide topological sphere theorems for compact submanifolds of spheres and Euclidean spaces, provided that the full norm of the second fundamental form is suitably bounded.     

A genus-4 topological recursion relation for Gromov-Witten invariants

In this paper,we give a new genus-4 topological recursion relation for Gromov-Witten invariants of compact symplectic manifolds via Pixton’s relations on the moduli space of curves.As an application,we prove that Pixton’s relations imply a known topological recursion relation on Mg,1 for genus g≤4.     

In response to the article by Maxim Kontsevich: Rozansky–Witten Invariants Via Formal Geometry

We show that recently constructed invariants of 3-dimensional manifolds and of hyperkähler manifolds (L. Rozansky and E. witten) come from characteristic classes of foliations and from Gelfand-fuks cohomology.