Existence of asymptotically hyperbolic Einstein 4-manifolds with T^3 cusps and nonzero signature

Determine whether there exists a complete, finite-volume Riemannian 4-manifold whose cusp ends are all diffeomorphic to T^3 × [0,∞), that admits an asymptotically hyperbolic Einstein metric, and whose signature σ is nonzero.

Background

Known constructions of finite-volume asymptotically hyperbolic Einstein 4-manifolds with T3 cusp ends include complete hyperbolic 4-manifolds and those obtained via Anderson’s generalized Dehn filling. In these families, the signature is necessarily zero (e.g., by the result of Long–Reid for hyperbolic 4-manifolds with T3 cusps).

The paper points out that, despite these constructions, it remains unsettled whether any asymptotically hyperbolic Einstein 4-manifold with only T3 cusp ends can have nonzero signature. Establishing an example or proving impossibility would clarify the range of topological types compatible with such Einstein geometries in the finite-volume cusp setting.

References

Note that all such examples are necessarily signature 0; to the best of the author's knowledge, it is still unknown whether there exist any manifolds admitting asymptotically hyperbolic Einstein metrics with only T3 cusps and nonzero signature.

— Seiberg-Witten Equations and Einstein Metrics on Finite Volume 4-Manifolds with Asymptotically Hyperbolic Ends  (2402.10366 - Xu, 2024) in Section 1 (Introduction)