Pinching surface groups in complex hyperbolic plane
We construct first examples of discrete geometrically finite subgroups of PU(2,1) which contain parabolic elements, and are isomorphic to surface groups.
math.DG↗
arXiv subjects
Igor Belegradek
Publications and source records attributed to Igor Belegradek.
Metrics of positive Ricci curvature on vector bundles over nilmanifolds
We construct metrics of positive Ricci curvature on some vector bundles over tori (or more generally, over nilmanifolds). This gives rise to the first examples of manifolds with positive Ricci curvature which are homotopy equivalent but not homeomorphic to manifolds of nonnegative sectional curvature.
math.DG↗
On Mostow rigidity for variable negative curvature
We prove a finiteness theorem for the class of complete finite volume Riemannian manifolds with pinched negative sectional curvature, fixed fundamental group, and of dimension $>2$. One of the key ingredients is that the fundamental group of such a manifold does not admit a small nontrivial action on an $R$-tree.
math.DG↗
Finiteness theorems for nonnegatively curved vector bundles
We prove several finiteness theorems for the normal bundles to souls in nonnegatively curved manifolds. More generally, we obtain finiteness results for open Riemannian manifolds whose topology is concentrated on compact domains of ``bounded geometry''.
math.DG↗
Topological obstructions to nonnegative curvature
We find new obstructions to the existence of complete Riemannian metric of nonnegative sectional curvature on manifolds with infinite fundamental groups. In particular, we construct many examples of vector bundles whose total spaces admit no nonnegatively curved metric.
math.DG↗
Pinching, Pontrjagin classes, and negatively curved vector bundles
We prove several finiteness results for the class $M_{a,b,G,n}$ of $n$-manifolds that have fundamental groups isomorphic to $G$ and that can be given complete Riemannian metrics of sectional curvatures within $[a,b]$ where $a\le b<0$. In particular, if $M$ is a closed negatively curved manifold of dimension at least three, then only finitely many manifolds in the class $M_{a,b,π_1(M), n}$ are total spaces of vector bundles over $M$. Furthermore, given a word-hyperbolic group $G$ and an integer $n$ there exists a positive $ε=ε(n,G)$ such that the tangent bundle of any manifold in the class $M_{-1-ε, -1, G, n}$ has zero rational Pontrjagin classes.
math.DG↗
Counting open negatively curved manifolds up to tangential homotopy equivalence
Under mild assumptions on a group G, we prove that the class of complete Riemannian n-manifolds of uniformly bounded negative sectional curvatures and with the fundamental groups isomorphic to G breaks into finitely many tangential homotopy types. It follows that many aspherical manifolds do not admit complete negatively curved metrics with prescribed curvature bounds.
math.DG↗
Intersections in hyperbolic manifolds
We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total spaces of orientable vector bundles over M.
math.GT↗