Realizability of cones over positively curved manifolds as cones at infinity

Establish whether every metric cone over an (n-1)-dimensional Riemannian manifold with sectional curvature at least 1 can occur as the cone at infinity of a complete, noncompact, n-dimensional Riemannian manifold with non-negative sectional curvature.

Background

The paper studies estimates relating scalar-curvature integrals to asymptotic volume ratios for complete noncompact Riemannian manifolds with non-negative sectional curvature. A central comparison object is the metric cone C(\Sigma) over an (n-1)-dimensional cross-section \Sigma with sectional curvature at least 1. The paper proves a scalar-curvature estimate directly for cones over smooth Riemannian manifolds, while separately proving an estimate for manifolds whose cones at infinity arise from asymptotic geometry.

The authors explain that the direct cone theorem cannot simply be deduced from the manifold theorem unless every cone C(\Sigma) over a Riemannian manifold \Sigma is realizable as the cone at infinity of a complete noncompact manifold with non-negative sectional curvature. Determining this realizability would clarify the relationship between the two theorems and the extent to which the cone-at-infinity result subsumes the direct cone result.