Contractible Positive-Scalar-Curvature Three-Manifolds

Definitions and mathematical statement

Interpret a complete contractible three-manifold with positive scalar curvature as a smooth Riemannian manifold $(M,g)$ of dimension $3$, using the usual convention that manifolds have no boundary. This boundary convention is not explicit in the supplied statement and requires confirmation. Completeness means completeness of the Riemannian distance induced by $g$. Contractibility means that there exist $p\in M$ and a continuous map $H:M\times[0,1]\to M$ such that \[ H(x,0)=x,\qquad H(x,1)=p\qquad(x\in M). \] Positive scalar curvature means \[ \operatorname{Scal}_g(x)>0\qquad\text{for every }x\in M; \] no uniform positive lower bound is assumed.

Question. For every such $(M,g)$, must there exist a homeomorphism \[ f:M\longrightarrow{\mathbb{R}}^3? \] The conclusion concerns the topology of $M$. No assumption of trivial fundamental group at infinity, bounded geometry, or curvature decay is included in the question. The supplied dated curation mentions results under additional hypotheses of these kinds, without specifying their precise formulations.

Short English statement

Must every contractible three-manifold admitting a complete Riemannian metric whose scalar curvature is positive at every point be homeomorphic to ${\mathbb{R}}^3$? The formulation uses the usual convention of smooth manifolds without boundary, which the source does not explicitly state. The supplied record labels the unrestricted question open.

Sources

• Local UnsolvedMath registry: supplied record with permanent ID $30004727$, titled Contractible Positive-Scalar-Curvature Three-Manifolds, in the Oberwolfach Reports Open Problems collection; exact source problem code catalogue entry. The record gives proposed year $2021$ and names no individual proposer.
• Supplied report citation: Analysis, Geometry and Topology of Positive Scalar Curvature Metrics (2022), Oberwolfach Reports, pp. 1579--1649, DOI 10.4171/owr/2021/30. https://doi.org/10.4171/owr/2021/30. The supplied background describes these reports as documenting mathematical workshops and their open-problem discussions.
• Supplied literature reference: Wang, Contractible three-manifolds and positive scalar curvature II. https://ems.press/journals/jems/articles/11806807.
• Supplied literature reference: Complete three-manifolds with quadratic decay. https://link.springer.com/article/10.1007/s00208-025-03192-9.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the question open and reports proofs under additional hypotheses such as trivial fundamental group at infinity, bounded geometry, or curvature decay. This is supplied dated curation derived from the OWR material, not an independently verified literature assessment; its status and cited claims require independent verification.
• The registry supplies the statement-status label corrected_verified, with the explanation that Euclidean-space notation was normalized without changing the topological question. This is a supplied editorial label, not independent mathematical verification.