Let 2≤ n<N, and let F:B^n→B^N be a proper holomorphic map admitting a C^N-n+1-smooth extension to the boundary. If the CR Gauss map of the boundary restriction F|_partialB^n is generically degenerate, must there exist automorphisms phi∈operatornameAut(B^n) and Psi∈operatornameAut(B^N), and a positive integer m, such that Psi circ F circphi=V_m oplus 0, where V_m(z)=(√fracm!alpha!z^alpha)_|alpha|=m is the degree-m Veronese map and the remaining target components are zero?

Xiaojun Huang’s rigidity problem for proper holomorphic ball maps

Let $2\le n<N$ , and let $F:\mathbb{B}^n\to\mathbb{B}^N$ be a proper holomorphic map admitting a $C^{N-n+1}$ -smooth extension to the boundary. If the CR Gauss map of the boundary restriction $F|_{\partial\mathbb{B}^n}$ is generically degenerate, must there exist automorphisms $\phi\in\operatorname{Aut}(\mathbb{B}^n)$ and $\Psi\in\operatorname{Aut}(\mathbb{B}^N)$ , and a positive integer $m$ , such that $\Psi\circ F\circ\phi=V_m\oplus 0$ , where $V_m(z)=\left(\sqrt{\frac{m!}{\alpha!}}z^\alpha\right)_{|\alpha|=m}$ is the degree- $m$ Veronese map and the remaining target components are zero?

References

Primary source

arXiv

Additional references

Rigidity for proper holomorphic ball maps with degenerate CR Gauss map — arXiv — Tianzhi Hu, Wanke Yin, Pingsan Yuan

Posted 1 day ago · edited 1 day ago

Progress summary

Refreshed 1 day ago

Claimed solved

A new specialist preprint claims a full classification of these maps, but the claim has not been independently checked.

Huang’s problem asks whether the relevant proper holomorphic maps between complex balls are, after automorphisms, Veronese maps with zero components.

Known results

Cima and Suffridge conjectured total geodesy below the boundary dimension $N=2n-1$ .
Huang, 1999: for $n>1$ and $N<2n-1$ , a sufficiently smooth map is equivalent to $(z_1,\ldots,z_n,0,\ldots,0)$ .
Nonlinear proper polynomial maps exist at $N=2n-1$ , including the Whitney map.

October 2026 claimed classification

Tianzhi Hu, Wanke Yin, and Pingsan Yuan’s preprint claims that the maps are equivalent to a Veronese map with zero components. This would settle the stated rigidity problem, but the preprint is unrefereed and the retrieved record contains no independent substantive assessment.

Current status (as of October 2026): Huang’s lower-dimensional linearity theorem is established, while the broader Veronese classification is only claimed in an unverified preprint.

Sources

No solutions have been posted yet.