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A proof of Gromov's cube inequality on scalar curvature

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arxiv 2105.12054 v4 pith:KITTRM22 submitted 2021-05-25 math.DG

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keywords cubeinequalitygromovmethodcurvaturedimensionminimalproof
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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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  1. Gap phenomenon for scalar curvature

    math.DG 2025-01 conditional novelty 6.0 of 10

    Scalar curvature on any closed even-dimensional manifold with nonzero Euler characteristic can be increased by at most an explicit constant, the gap, which is a function of the minimal eigenvalue of the curvature oper...

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