On CR maps from the sphere into the tube over the future light cone

Abstract

We determine all local smooth or formal CR maps from the unit sphere S3⊂C2 into the tube T:=C×iR3⊂C3 over the future light cone C:={x∈R3:x12+x22=x32,x3>0}. This result leads to a complete classification of proper holomorphic maps from the unit ball in C2 into Cartan's classical domain of type IV in C3 that extend smoothly to some boundary point. Up to composing with CR automorphisms of the source and target, the classification consists of four algebraic maps. Two maps among them were known earlier in the literature. They can be generalized to higher dimensional cases and were shown to be “rigid” when the source dimension is at least 4 in a recent paper by Xiao and Yuan. Two newly discovered quadratic polynomial maps provide counterexamples to a conjecture appeared in the same paper for the case of dimension two.