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Macroscopic scalar curvature and codimension 2 width

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arxiv 2112.04594 v1 pith:MK5VDO4D submitted 2021-12-08 math.DG math.MG

Hannah Alpert , Alexey Balitskiy , Larry Guth This is my paper
classification math.DGmath.MG
keywords balleverymacroscopicradiuscurvatureassumptionscodimensioncomplete
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abstract

We show that a complete $3$-dimensional Riemannian manifold $M$ with finitely generated first homology has macroscopic dimension $1$ if it satisfies the following "macroscopic curvature" assumptions: every ball of radius $10$ in $M$ has volume at most $4$, and every loop in every ball of radius $1$ in $M$ is null-homologous in the concentric ball of radius $2$.

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  1. 1-Uryson width and covers

    math.MG 2025-05 conditional novelty 7.0 of 10

    Compact surfaces satisfy UW1(Σ) ≤ UW1(tilde Σ), virtually cyclic polyhedra satisfy UW1(X) ≤ 6 UW1(tilde X), and any counterexample to the width question reduces to 2-complexes.

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