Nonnegative-scalar-curvature contractible 3-manifold conjecture

Prove that every complete, connected, open, contractible 3-manifold with nonnegative scalar curvature is diffeomorphic to \(\mathbb{R}^3\), without assuming local conformal flatness, bounded geometry, or other additional hypotheses.

Background

A longstanding conjecture discussed in the paper asserts Euclidean rigidity for complete, open, contractible 3-manifolds with nonnegative scalar curvature. Earlier results establish the conclusion under stronger assumptions, such as uniformly positive scalar curvature with finitely generated fundamental group, and more recent work verifies it under bounded geometry.

The paper proves the conjecture in the locally conformally flat setting by showing that the developing image has a one-point conformal boundary and is therefore conformally equivalent to Euclidean space. The unrestricted conjecture for arbitrary complete metrics with nonnegative scalar curvature remains unresolved.

References

Motivated by the work of Gromov--Lawson, the following conjecture has circulated in the field for decades, but has only recently begun to receive attention: A complete, connected, open, contractible $3$-manifold with nonnegative scalar curvature is diffeomorphic to $\mathbb{R}3$.

— Locally Conformally Flat Manifolds with Positive Scalar Curvature: Kleinian Groups, Moduli Spaces, and Euclidean Rigidity  (2609.02267 - Deng, 2 Sep 2026) in Introduction, subsection on complete noncompact manifolds; reiterated in Section 6.1