BENDING DEFORMATIONS OF COMPLEX HYPERBOLIC SURFACES

BORIS APANASOV

Abstract. 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 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 lowdimensional 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.