Drawstrings and flexibility in the Geroch conjecture

TL;DR

This paper constructs new examples of 3-manifolds with lower scalar curvature bounds that challenge existing conjectures on the stability of the Geroch Conjecture, advancing understanding of geometric stability and structure.

Contribution

It provides the first counterexample to Sormani’s conjecture on Geroch Conjecture stability and establishes a $W^{1,p}$-stability result for warped product manifolds.

Findings

Counterexample to Sormani's conjecture on Geroch Conjecture stability

Construction of warped-product manifolds with almost nonnegative scalar curvature

Validation of $W^{1,p}$-stability in warped product class

Abstract

In this paper, we observe new phenomena related to the structure of 3-manifolds satisfying lower scalar curvature bounds. We construct warped-product manifolds of almost nonnegative scalar curvature that converge to pulled string spaces in the Sormani-Wenger intrinsic flat topology. These examples extend the results of Lee-Naber-Neumayer \cite{LNN} to the case of dimension $3$. As a consequence, we produce the first counterexample to a conjecture of Sormani \cite{SormaniConj} on the stability of the Geroch Conjecture. Our example tests the appropriate hypothesis for a related conjecture of Gromov. On the other hand, we demonstrate a $W^{1,p}$-stability statement ($1\leq p<2$) for the Geroch Conjecture in the class of warped products.