In their celebrated paper of 1939 Myers and Steenrod showed that
the group of isometries of a Riemannian manifold acts properly on the
manifold. This fact has many consequences. In particular, it
implies that the group of isometries is a Lie group in the compactopen
topology. This result triggered extensive studies of closed subgroups of
the isometry groups of Riemannian manifolds. The peak of activities in
this area occurred in the 1950s70s, with many outstanding
mathematicians involved: Kobayashi, Nagano, Yano, H.C. Wang, I. P. Egorov,
to
name a few. In particular, Riemannian manifolds whose isometry groups
possess subgroups of sufficiently high dimensions were explicitly
determined.
I will speak about proper actions in the complexgeometric setting. In
this setting (real) Lie groups act properly by holomorphic transformations
on complex manifolds. My general aim is to build a theory parallel to the
theory that exists in the Riemannian case. In my lecture I will survey
recent classification results for complex manifolds that admit proper
actions of highdimensional groups.
