Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds

Ervin Hadziosmanovic

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arXiv · 2507.20247

Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds

Ervin Hadziosmanovic

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Abstract

We prove that for manifolds with negative curvature bounded away from $0$ of infinite volume and bounded geometry, the bounded fundamental class, defined via integration of the volume form over straight top-dimensional simplices, vanishes if and only if the Cheeger isoperimetric constant is positive. This gives a partial affirmative answer to a conjecture of Kim and Kim. Furthermore, we show that for all manifolds with negative curvature bounded away from $0$ of infinite volume, the positivity of the Cheeger constant implies the vanishing of the bounded volume class, solving one direction of the conjecture in full generality.

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BibTeX RIS

Ervin Hadziosmanovic. 2025-07-27. Bounded volume class and Cheeger isoperimetric constant for negatively curved manifolds. https://arxiv.org/abs/2507.20247

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