Hyperbolic 2-Dimensional Manifolds with 3-Dimensional Automorphism Group

If M is a connected n-dimensional Kobayashi-hyperbolic complex manifold, then the group Aut(M) of holomorphic automorphisms of M is a (real) Lie group in the compact-open topology, of dimension d(M) not exceeding n + 2n, with the maximal value occurring only for manifolds holomorphically equivalent to the unit ball B ⊂ C [Ko1], [Ka]. We are interested in describing hyperbolic manifolds with lower (but still sufficiently high) values of d(M). The classification problem for hyperbolic manifolds with highdimensional automorphism group is a complex-geometric analogue of that for Riemannian manifolds with high-dimensional isometry group, which inspired many results in the 1950’s-70’s (see [Ko2] for details). The principal underlying property that made the classification in the Riemannian case possible is that the group of isometries acts properly on the manifold – see [MS], [vDvdW] (a topological group G is said to act properly on a manifold S if the map G×S → S×S, (g, p) 7→ (gp, p) is proper). In the case of hyperbolic manifolds, the action of the group Aut(M) is proper as well (see [Ko1], [Ka]), and, as in the Riemannian case, this property is critical for our arguments, despite the fact that our techniques are almost entirely different from those utilized for isometry groups. In [IKra], [I1] we completely classified manifolds with n ≤ d(M) < n+2n (partial classifications for d(M) = n were also obtained in [GIK] and [KV]). Note that for d(M) = n the manifold M may not be homogeneous,