Classification of complete Chern-flat metrics on \(\mathbb{C}^2\)

Determine whether every complete Chern-flat Hermitian metric on \(\mathbb{C}^2\) is holomorphically isometric to a metric of the displayed \((\lambda,\mu)\)-family, equivalently whether the metrics obtained from \(\omega_0=\sqrt{-1}(dz_1\wedge d\overline{z}_1+e^{2\operatorname{Re}( ho(z_1))}dz_2\wedge d\overline{z}_2)\) by the specified coframe construction exhaust all complete Chern-flat Hermitian metrics on \(\mathbb{C}^2\).

Background

The paper studies complete Hermitian manifolds with vanishing Chern curvature, or complete Chern-flat Hermitian manifolds. Its main uniformization theorem shows that polynomial growth of the Chern torsion forces the universal cover to be biholomorphic to a product of a complex Euclidean space and a simply connected complex Lie group, but the authors emphasize that this does not classify all complete Chern-flat metrics or imply holomorphic isometry to a left-invariant metric.

For C2\mathbb{C}^2, the authors introduce a specific family of complete Chern-flat metrics determined by an entire function ρ\rho and constants λ,μ\lambda,\mu. The unresolved problem asks whether every complete Chern-flat Hermitian metric on C2\mathbb{C}^2 is holomorphically isometric to one of these metrics, thereby giving a complete classification in complex dimension two.

References

Is any complete Chern-flat Hermitian metric on $\mathbb{C}2$ holomorphically isometric to

\omega= \sqrt{-1} ( \varphi1\wedge \overline{\varphi1} + \varphi2 \wedge \overline{\varphi2} )

Here $\varphi1=dz_1+(\lambda\,z_1+\mu)\,e{\rho(z_1)}dz_2$, $\varphi2=e{\rho(z_1)}dz_2$ for some $\rho\in\mathcal{O}(\mathbb{C})$ and constants $\lambda, \mu \in \mathbb{C}$. In other words, consider the following complete Chern-flat Hermitian metric

\omega_0= \sqrt{-1} ( dz_1\wedge d\overline{z}_1 + e{2\operatorname{Re}(\rho(z_1))}dz_2 \wedge d\overline{z}_2 ).

We expect that the $(\lambda, \mu)$-family of Hermitian metrics constructed from $\omega_0$ in the sense of C2Herm_intro_2 exhausts all complete Chern-flat Hermitian metrics on $\mathbb{C}2$.

— Holomorphic functions on complete Hermitian manifolds with flat Chern connection, II  (2609.10159 - Li et al., 9 Sep 2026) in Section 1, subsection “Further discussion,” Question \ref{ques_intro}