A proof of Gromov's cube inequality on scalar curvature

Jinmin Wang; Zhizhang Xie; Guoliang Yu

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arXiv · 2105.12054

A proof of Gromov's cube inequality on scalar curvature

Jinmin Wang·Zhizhang Xie·Guoliang Yu

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Abstract

Gromov proved a cube inequality on the bound of distances between opposite faces of a cube equipped with a positive scalar curvature metric in dimension $\leq 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\geq 9$. In this paper, we prove Gromov's cube inequality in all dimensions with the optimal constant via Dirac operator method. In fact, our proof yields a strengthened version of Gromov's cube inequality, which does not seem to be accessible by minimal surface method.

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Jinmin Wang, Zhizhang Xie, Guoliang Yu. 2021-05-25. A proof of Gromov's cube inequality on scalar curvature. https://arxiv.org/abs/2105.12054

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