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arXiv · 2309.03756

Drawstrings and flexibility in the Geroch conjecture

Demetre Kazaras·Kai Xu
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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.

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Demetre Kazaras, Kai Xu. 2023-09-07. Drawstrings and flexibility in the Geroch conjecture. https://arxiv.org/abs/2309.03756

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