  We show that there does not exist a Kobayashi hyperbolic complex manifold of
dimension $n\ne 3$, whose group of holomorphic automorphisms has dimension
$n^2+1$ and that, if a 3-dimensional connected hyperbolic complex manifold has
automorphism group of dimension 10, then it is holomorphically equivalent to
the Siegel space. These results complement earlier theorems of the authors on
the possible dimensions of automorphism groups of domains in comlex space. The
paper also contains a proof of our earlier result on characterizing
$n$-dimensional hyperbolic complex manifolds with automorphism groups of
dimension $\ge n^2+2$.
