differential geometry

Drawstrings and flexibility in the Geroch conjecture

arXiv:2309.03756

summary

The paper constructs warped‑product 3‑manifolds with almost nonnegative scalar curvature that converge to pulled‑string spaces, providing a counterexample to a stability conjecture for the Geroch conjecture and proving a W^{1,p} stability result for warped products.

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 . 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 -stability statement () for the Geroch Conjecture in the class of warped products.

Minor changes and improved exposition, final version accepted for publication

Topics & keywords

#scalar curvature#3-manifolds#warped products#intrinsic flat convergence#Geroch conjecturewarped-product manifoldsalmost nonnegative scalar curvatureSormani‑Wenger intrinsic flat topologypulled string spacesW^{1,p} stability
Drawstrings and flexibility in the Geroch conjecture · wovepaper