Bending deformations of complex hyperbolic surfaces

We study deformations of complex hyperbolic surfaces which furnish the simplest examples of: (i) negatively curved K\"ahler 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.