Rigidity of Bergman-isometric component maps on smooth bounded domains

Determine whether, for every smooth bounded domain D in C^n with n≥2 and every connected open subset U⊂D, holomorphic maps f_1,…,f_m:U→D of full rank at some point each, satisfying ∑_{j=1}^m f_j^*ω_D=λω_D for some λ>0, necessarily extend individually to elements of Aut(D).

Background

The paper proves this componentwise rigidity under substantially stronger hypotheses: D is simply connected, bounded, strongly pseudoconvex, and has real-analytic boundary. Under those assumptions, every component extends to an automorphism and λ=m.

The question asks whether the same conclusion remains valid for arbitrary smooth bounded domains of complex dimension at least two. The authors note that nonstandard one-dimensional and polydisc examples prevent an unrestricted statement in dimension one, motivating the dimensional restriction.

References

Nevertheless, we conjecture that the following rigidity result holds when the dimension is at least 2.

Let $D$ be a smooth bounded domain in $n, n \geq 2$ and $U \subset D$ be a connected open subset. For $1\leq j\leq m$, let $ f_j: U\rightarrow D$ be a holomorphic map, and assume that every $f_j$ has full rank $n$ at some point of $U$. If there exists $\lambda>0$ such that \begin{equation*} \sum_{j=1}{m}f_j*\omega_D=\lambda\omega_D \end{equation*} holds on $U$, does each $ f_j$ extend to an element in $Aut(D)$?

— Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem  (2608.15991 - Yuan, 17 Aug 2026) in Section 6.1, Question

In contrast, no analogous result is currently known when $p=n$.

— Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem  (2608.15991 - Yuan, 17 Aug 2026) in Section 7, On holomorphic maps preserving Bergman (p,p)-forms

Moreover, it remains open whether the conclusion of Theorem~\ref{product}, together with its subsequent consequences, continues to hold under the analogous preservation condition for Bergman $(p,p)$-forms with $p>1$.

— Holomorphic Bergman isometries between bounded domains and the extension of Lu's theorem  (2608.15991 - Yuan, 17 Aug 2026) in Section 7, On holomorphic maps preserving Bergman (p,p)-forms