arXiv subjects
Publications and source records attributed to Boris Apanasov.
We develop and study quaternionic and octonionic analogies of Cartan angular and Toledo invariants that are well known in the complex hyperbolic space. Using such invariants we study quasifuchsian deformations (including bendings) of quaternionic and octonionic hyperbolic manifolds.
We describe relations between hyperbolic geometry and codimension two knots or, more exactly, between varieties of conjugacy classes of discrete faithful representations of the fundamental groups of hyperbolic n-manifolds M into $\operatorname{SO}^{\circ} (n+2,1)$ and (n-1)-dimensional knots in the (n+1)-sphere. This approach allows us to discover a phenomenon of non-connectedness of these varieties for closed n-manifolds M, $n\geq 3$, with large enough number of disjoint totally geodesic surfaces, to construct quasisymmetric infinitely compounded "Julia" knots $K\subset S^{n+1}$ which are everywhere wild and have recurrent $π_1(M)$-action, and to study circle and 2-plane bundles (with geometric structures) over closed hyperbolic n-manifolds.
This paper concerns with deformations of noncompact complex hyperbolic manifolds (with locally Bergman metric), varieties of discrete representations of their fundamental groups into $PU(n,1)$ and the problem of (quasiconformal) stability of deformations of such groups and manifolds in the sense of L.Bers and D.Sullivan. Despite Goldman-Millson-Yue rigidity results for such complex manifolds of infinite volume, we present different classes of such manifolds that allow non-trivial (quasi-Fuchsian) deformations and point out that such flexible manifolds have a common feature being Stein spaces. While deformations of complex surfaces from our first class are induced by quasiconformal homeomorphisms, non-rigid complex surfaces (homotopy equivalent to their complex analytic submanifolds) from another class are quasiconformally unstable, but nevertheless their deformations are induced by homeomorphisms.
We study geometry, topology and deformation spaces of noncompact complex hyperbolic manifolds (geometrically finite, with variable negative curvature), whose properties make them surprisingly different from real hyperbolic manifolds with constant negative curvature. This study uses an interaction between Kähler geometry of the complex hyperbolic space and the contact structure at its infinity (the one-point compactification of the Heisenberg group), in particular an established structural theorem for discrete group actions on nilpotent Lie groups.
We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved Kähler manifolds and (ii) negatively curved Riemannian manifolds not having {\it constant} curvature. Although such complex surfaces may share the rigidity of quaternionic/octionic hyperbolic manifolds, our main goal is to show that they enjoy nevertheless the flexibility of low-dimensional real hyperbolic manifolds. Namely we define a class of ``bending" deformations of a given (Stein) complex surface $M$ associated with its closed geodesics provided that $M$ is homotopy equivalent to a Riemann surface whose embedding in $M$ has a non-trivial totally real geodesic part. Such bending deformations bend $M$ along its closed geodesics and are induced by equivariant quasiconformal homeomorphisms of the complex hyperbolic space and its Cauchy-Riemannian structure at infinity.
The paper sketches a recent progress and formulates several open problems in studying equivariant quasiconformal and quasisymmetric homeomorphisms in negatively curved spaces as well as geometry and topology of noncompact geometrically finite negatively curved manifolds and their boundaries at infinity having Carnot--Carathéodory structures. Especially, the most interesting are complex hyperbolic manifolds with Cauchy--Riemannian structure at infinity, which occupy a distinguished niche and whose properties make them surprisingly different from real hyperbolic ones.