What question did this study set out to answer?

October 5, 2026Open Access

Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov

View Full Paper

Ask AI

Bookmark

Test the validity of Gromov's conjecture proposing that complete Riemannian manifolds with nonnegative scalar curvature that are isometric at infinity to flat space must be entirely flat.
Constructed an explicit complete Riemannian metric on R² × S^{n−2} parameterized by a single smooth step function.
Employed the geometric mechanism of Witten's Kaluza–Klein bubble alongside cut-and-paste and quotient techniques to generalize the metrics to flat ends of the form T^k × R^N and T^{n−1} × R.
Demonstrated that the constructed metrics possess nonnegative scalar curvature globally, strictly positive scalar curvature on an open set, and are isometric to flat space outside a compact set for all dimensions n ≥ 3.
Proved that the counterexample manifolds are simply connected and spin for all dimensions n ≥ 4, directly disproving Gromov's conjecture.

Discussion

No takes yet. Share an insight, caveat, or question.

Implication

Geometric construction demonstrates nonflat Riemannian manifolds with nonnegative scalar curvature that are flat at infinity for dimensions n ≥ 3, highlighting the failure of Gromov's conjecture.

Cite This Study

Alper Ferudun (2026) studied this question.

synapsesocial.com/papers/6ac33da06931ac3d2d78d789 https://doi.org/10.5281/zenodo.23121035

View Full Paper

Ask AI

Bookmark

Implication

Geometric construction demonstrates nonflat Riemannian manifolds with nonnegative scalar curvature that are flat at infinity for dimensions n ≥ 3, highlighting the failure of Gromov's conjecture.

Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov

Key Points

Discussion

Implication

Cite This Study

Nonflat Metrics with Nonnegative Scalar Curvature That Are Flat at Infinity: A Counterexample to a Conjecture of Gromov

Key Points

Discussion

Implication

Cite This Study

Also Consider

Also Consider