Invariant proper holomorphic maps between balls

[摘要] We consider proper holomorphic maps between balls that are invariant under the action of finite groups of unitary matrices. We are primarily interested in actions of groups that are fixed-point-free; for purposes of comparison we will briefly consider matrix groups that act with fixed points (that is, groups that have at least one nontrivial element with an eigenvalue of one) in the last chapter. Forstneric showed that given any finite unitary fixed-point-free matrix group, there exists a proper holomorphic map from the ball in the appropriate dimensional complex Euclidean space to a higher dimensional ball, that is invariant under the action of that group. He showed on the other hand that if we also require the map to be smooth to the boundary, then many groups are ruled out.One of our main results is the following theorem: if f is a proper holomorphic map between balls that is invariant under the action of some finite fixed-point-free matrix subgroup of a unitary group (acting on the domain of f), and, in addition, smooth to the boundary, then necessarily that group is cyclic. We rule out some of these cyclic unitary groups as well. We give corollaries concerning the nonexistence of smooth CR mappings from certain spherical space forms to spheres.We next prove some propositions related to the theory of polynomial proper mappings between balls. As another important result, in cases where there are known finite fixed-point-free matrix group-invariant mappings we classify all such maps in terms of a group-basic map. In a subsequent chapter we investigate existence and nonexistence of various sorts of polynomial proper maps between balls, mostly invariant under some matrix group action, from a combinatorial perspective. We give a simple means of depicting monomial mappings from the ball in two-dimensional space, and show some applications. As a final theorem, we show how proper holomorphic maps between balls, invariant under the action of finite matrix groups possibly acting with fixed points, can be "constructed". This uses a technique developed by Low. We derive some interesting examples from this construction.

[关键词] Mathematics