Fundamental-group instability for positive Einstein four-manifolds

Determine whether fundamental-group instability can occur among closed Einstein four-manifolds with positive Einstein constant, including cases with limiting geometries other than the spherical degeneration excluded by the Huang–Ozuch isolation theorem.

Background

The paper constructs pairs of families of closed Ricci-flat Kähler four-manifolds with non-isomorphic fundamental groups that converge in the Gromov–Hausdorff sense to the same compact flat orbifold. It then asks whether an analogous instability phenomenon can occur for Einstein metrics with positive Einstein constant.

The spherical degeneration considered earlier is ruled out by the Huang–Ozuch isolation theorem, which prevents spherical and hyperbolic Einstein four-orbifolds with isolated singularities from being approximated by smooth Einstein four-manifolds. The unresolved problem is therefore whether another limiting geometry could support fundamental-group instability in the positive-Einstein setting.

References

Several natural questions remain. First, can fundamental-group instability occur among closed Einstein four-manifolds with positive Einstein constant? Although the isolation theorem of Huang--Ozuch rules out the spherical degeneration discussed above, the question remains open for other possible limiting geometries.

— Instability of the Fundamental Group for Ricci-Flat Kähler 4-Manifolds  (2610.05795 - Bellati et al., 5 Oct 2026) in Section 1, Introduction, paragraph beginning “Several natural questions remain.”