Higher-dimensional low-regularity sphericity theorem

Establish whether a rigid, strictly pseudoconvex CR hypersurface of class C^4 in higher complex dimensions is spherical whenever its fundamental Chern–Moser invariant vanishes identically, as asserted by the authors’ belief that the low-regularity result of Kossovskiy and Zaitsev extends beyond complex dimension two.

Background

The paper uses the correspondence between Bochner-flat Kähler manifolds and spherical Sasakian manifolds. To infer sphericality from vanishing of the fundamental Chern–Moser invariant, the authors discuss regularity requirements for strictly pseudoconvex CR hypersurfaces. They cite a result of Kossovskiy and Zaitsev establishing this implication for hypersurfaces in complex dimension two of class C6.

The authors state that the analogous implication is believed to hold in higher dimensions for rigid, strictly pseudoconvex CR hypersurfaces of class C4. Such an extension would apply to the rigid CR hypersurfaces arising from their Kähler–Sasakian correspondence and would lower the a priori smoothness required of the Kähler metric to class C2.

References

This result is believed to remain true in higher dimensions for rigid, strictly pseudoconvex CR hypersurfaces of class $\mathcal C4$, which applies to our situation and reduces the required a priori smoothness of the metric tensor to $\mathcal C2$.

— Canonical Bochner-Kähler potentials via the Sasaki-Kähler correspondence  (2609.03791 - Kolář et al., 3 Sep 2026) in Remark following Proposition 2, Section 2, “Metric Curvature and CR curvature”