A proof of Gromov's cube inequality on scalar curvature

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 $\le 8$ using minimal surface method. He conjectured that the cube inequality also holds in dimension $\ge 9$. 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.

21 pages. v3 to v4: the proof for the strict inequality has been revised

References in corpus (2)

• No metrics with Positive Scalar Curvatures on Aspherical 5-Manifolds
• A quantitative relative index theorem and Gromov's conjectures on positive scalar curvature

Cited by in corpus (3)

• The positive mass theorem and distance estimates in the spin setting
• Band width and the Rosenberg index
• On Gromov's dihedral extremality and rigidity conjectures