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

TL;DR

This paper classifies all local smooth or formal CR maps from the 3-sphere into a tube over the future light cone, leading to a complete classification of certain proper holomorphic maps into Cartan's domain of type IV, revealing new quadratic polynomial examples.

Contribution

It provides a full classification of CR maps from the sphere to the tube over the light cone and identifies new quadratic polynomial maps that counter a previous conjecture.

Findings

Four algebraic maps classify the CR maps up to automorphisms.

Two maps were previously known and shown to be rigid.

Two new quadratic polynomial maps serve as counterexamples to a conjecture.

Abstract

We determine all local smooth or formal CR maps from the unit sphere $\mathbb{S}^3\subset \mathbb{C}^2$ into the tube $\mathcal{T}:= \mathcal{C} \times i\mathbb{R}^3 \subset \mathbb{C}^3$ over the future light cone $\mathcal{C}:= \left\{x\in \mathbb{R}^3\colon x_1^2+x_2^2 = x_3^2, \ x_3 > 0\right\}$. This result leads to a complete classification of proper holomorphic maps from the unit ball in $\mathbb{C}^2$ into Cartan's classical domain of type IV in $\mathbb{C}^3$ 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, which were shown to be ``rigid'' in the higher dimensional case 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.