Contractible Ricci-Nonnegative Four-Manifolds with Euclidean Growth

Definitions and mathematical statement

Let $(M^4,g)$ be a smooth, complete, contractible Riemannian four-manifold. Contractibility means that the identity map of $M$ is homotopic to a constant map. Write $d_g$ for the Riemannian distance and $\operatorname{Vol}_g$ for Riemannian volume. Nonnegative Ricci curvature means \[ \operatorname{Ric}_g(v,v)\geq 0\qquad\text{for every }x\in M\text{ and }v\in T_xM. \] Use the conventional meaning of Euclidean volume growth: \[ \liminf_{r\to\infty}\frac{\operatorname{Vol}_g(B_g(p,r))}{r^4}>0 \] for some base point $p\in M$. Here $B_g(p,r)$ is the metric ball of radius $r$. This growth condition is independent of the choice of the fixed base point.

Conjecture 1 (as supplied). Under these hypotheses, both of the following assertions hold:

1. $M$ is homeomorphic to ${\mathbb{R}}^4$.
2. For every $(Z,d_Z)\in\mathcal C_\infty$, the space $Z$ is homeomorphic to \[ S^3=\{x\in{\mathbb{R}}^4:\|x\|=1\}. \]

These are assertions of homeomorphism; no diffeomorphism conclusion is stated.

Unresolved definition of the second assertion. The supplied original passage uses $\mathcal C_\infty$ without defining it, and the cleaned statement calls its members asymptotic cross-sections. A conventional interpretation is the collection of cross-sections of all metric cones arising as tangent cones at infinity of $(M,d_g)$: a compact metric space $(Z,d_Z)$ belongs to this collection when, for some sequence $r_i\to\infty$, \[ (M,r_i^{-1}d_g,p)\longrightarrow (C(Z),d_C,o) \] in the pointed Gromov--Hausdorff sense. Here $C(Z)$ is the metric cone over $Z$, and $o$ is its vertex. Under this interpretation, assertion (ii) quantifies over every such cross-section and every such limiting sequence, without assuming uniqueness of the tangent cone. The identification of this conventional collection with the source's $\mathcal C_\infty$ requires confirmation from the surrounding report; it is an explicit interpretation, not a definition supplied by the excerpt.

Short English statement

The supplied conjecture asks whether every smooth, complete, contractible four-manifold with nonnegative Ricci curvature and Euclidean volume growth is homeomorphic to ${\mathbb{R}}^4$, and whether every asymptotic cross-section is homeomorphic to $S^3$. The source notation for the collection of cross-sections is undefined in the supplied excerpt, so the second assertion requires confirmation of its intended scope.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $30006421$, source problem code catalogue entry, titled "Contractible Ricci-Nonnegative Four-Manifolds with Euclidean Growth," in the Oberwolfach Reports Open Problems collection. The original excerpt labels the statement "Conjecture 1." No individual proposer is supplied; the proposed year is recorded as 2025.
• Supplied report citation: Differentialgeometrie im Großen (2026), Oberwolfach Reports, pp. 1591--1646, DOI 10.4171/owr/2025/30. https://doi.org/10.4171/owr/2025/30. The citation's 2026 title date and the DOI's 2025 component are retained as supplied.
• Supplied literature reference: Brué--Pigati--Semola survey. https://doi.org/10.1016/j.indag.2025.04.003.
• Supplied literature reference: Huang--Huang fundamental groups. https://arxiv.org/abs/2502.03259.
• Supplied literature reference: Oberwolfach Report 30. https://ems.press/content/serial-article-files/52249.
• Literature assessment dated 2026-08-22: the supplied curation labels the conjecture open, says that a 2026 survey still asks about the existence of a nonstandard contractible four-manifold with nonnegative Ricci curvature and Euclidean volume growth, and describes recent work as providing structural and fundamental-group restrictions only. This is supplied dated curation derived from the OWR record, not an independently verified literature assessment. The record explicitly requests independent verification of time-sensitive status and cited claims.