  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.
