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Macroscopic scalar curvature and codimension 2 width

· 2021 · math.DG · arXiv 2112.04594

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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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math.MG 1

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2025 1

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CONDITIONAL 1

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1-Uryson width and covers

math.MG · 2025-05-27 · conditional · novelty 7.0

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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  • 1-Uryson width and covers math.MG · 2025-05-27 · conditional · none · ref 1 · internal anchor

    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.