# Complexity of mappings in CR geometry

September 11 to September 15, 2006

at the

American Institute of Mathematics, Palo Alto, California

organized by

John P. D'Angelo and Peter Ebenfelt

## Original Announcement

This workshop will study mappings in CR geometry. Participants will focus on a quite specific part of this general subject, namely the relationship between the complexity of a CR mapping and the complexity of the CR structures on the domain and target manifolds.

The workshop has three principal goals. The first goal is to determine the fundamental notions of CR complexity and to prove sharp results about these notions. The second goal is to organize CR complexity theory into a broad framework that will be useful in CR geometry and also apply to other parts of mathematics. The third goal is to bring active senior researchers and young mathematicians together to work in a focused manner that will forge interactions and guide future research.

We illustrate the idea using the unit sphere, which is in some precise sense the simplest CR manifold. Consider a proper holomorphic mapping between balls in complex Euclidean spaces of possibly different dimensions. When the domain dimension is at least two, and the mapping is smooth at the boundary, then it must be a rational mapping. Its restriction to the boundary sphere defines a non-constant CR mapping between spheres. One may ask the following natural questions: How complicated can a CR mapping between spheres be, given the domain and target dimensions? What are the appropriate measures of complexity?

Let us first consider the equidimensional case. In one dimension we are considering finite Blaschke products, whose complexity can be measured by the number of factors, or equivalently, the degree of the divisor. In higher dimensions, it is a well-known result (Pinchuk and Alexander) that a proper holomorphic mapping between equidimensional balls is an automorphism, and hence a rational mapping of degree one. Moreover, the mapping is spherically equivalent to the identity mapping, and hence it is not complicated. When the target space has lower dimension than the domain space, the only CR mappings between spheres are constant, and thus even less complicated. On the other hand, as the dimension of the target space becomes large relative to that of the domain, the dimension of the moduli space of CR mappings between spheres is unbounded. One can create rational CR mappings between spheres of arbitrarily large degree (and arbitrary complexity in other senses) by allowing the target dimension to be large enough (D'Angelo).

On the other hand, in low codimension there are restrictions on the degree (Faran, Webster, Huang-Ji). Thus there is a relationship between the complexity of a rational CR mapping (as measured by the degree in this case) between spheres and its domain and target dimensions. Making this relationship quantitative and precise is a difficult and fundamental problem that, along with other related problems, will guide the workshop.

## Material from the workshop

A list of participants.

The workshop schedule.

A report on the workshop activities.

There is a short glossary of terms related to the workshop topic.