The classification of CR maps from hyperquadrics into tubes over null cones of symmetric forms
Authors
Nguyen Gia Hien
Michael Reiter
Duong Ngoc Son
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 Kähler 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).
