A Smooth Intrinsic Flat Limit of with Negative Curvature

TL;DR

This paper constructs a sequence of positively curved manifolds whose intrinsic flat limit exhibits negative scalar curvature, addressing a question about curvature preservation under convergence.

Contribution

It provides a counterexample showing that nonnegative scalar curvature is not necessarily preserved under intrinsic flat convergence.

Findings

Intrinsic flat limits can have negative scalar curvature

Counterexample to Gromov's question from 2014

Shows limits can differ significantly from approximating manifolds

Abstract

In 2014, Gromov asked if nonnegative scalar curvature is preserved under intrinsic flat convergence. Here we construct a sequence of closed oriented Riemannian $n$-manifolds, $n\geq 3$, with positive scalar curvature such that their intrinsic flat limit is a Riemannian manifold with negative scalar curvature.