The classification of CR maps from hyperquadrics into tubes over null cones of symmetric forms

Abstract

We classify CR maps from the hyperquadric of signature $l>0$ in $\mathbb{C}^n$, $n\geq 3$, to the local model for the tube over the null cone of a symmetric form in $\mathbb{C}^{n+1}$, up to CR automorphisms of the source and target. In contrast to the setting of the Heisenberg hypersurface in $\mathbb{C}^3$ (i.e., the case $l=0$), studied earlier in Reiter--Son [27], our analysis uncovers two new equivalence classes of CR maps of geometric rank one and one new class of geometric rank two in the case $n=3$. In the case $n\geq 4$, we establish that all maps extend to local isometries of certain indefinite K\"ahler metrics. We further derive a classification of (local) proper holomorphic maps from the generalized unit ball $\mathbb{B}^n_l$ into a generalized version of the Lie ball $D^{\mathrm{IV}}_{m,l}$ (the generalized classical domain of type~IV).