Existence of complete Chern-flat metrics on the smooth affine quadric

Determine whether the smooth affine quadric \(Q=\{z_1^2+z_2^2+z_3^2=1\}\subset\mathbb{C}^3\) admits a complete Chern-flat Hermitian metric.

Background

The paper observes that $\mathbb{C}^2\setminus\{0\}$ admits no complete Chern-flat Hermitian metric, despite being elliptic, because a hypothetical global unitary parallel holomorphic frame would extend across the puncture by Hartogs’ theorem and produce an impossible complete metric on $\mathbb{C}^2$.

The authors then turn to simply connected Stein surfaces with trivial canonical bundle, where the Hartogs-extension obstruction does not directly apply. They identify the smooth affine quadric $Q$ as an example and ask whether it supports a complete Chern-flat metric. They note that if such a metric exists, its torsion cannot have polynomial growth by the paper’s uniformization theorem.