Gromov's square inequality for spin manifolds with cube-like boundaries

Let $(X,g)$ be an $n$-dimensional compact connected orientable manifold with boundary, and let $\underline{X}_\bullet$ be a closed orientable manifold of dimension $n-m$. Suppose

$$f\colon X\to[-1,1]^m\times\underline{X}_\bullet$$

is a continuous map sending the boundary of $X$ to the boundary of $[-1,1]^m\times\underline{X}_\bullet$ and having non-zero degree. Let $\partial_{j\pm}$, $j=1,\dots,m$, be the pullbacks of the pairs of opposite faces of $[-1,1]^m$ under the composition with the projection to $[-1,1]^m$. Assume that for any $m$ hypersurfaces $Y_j\subset X$ separating $\partial_{j-}$ from $\partial_{j+}$, their transversal intersection $Y_\pitchfork$ does not admit a metric with positive scalar curvature, and neither do $Y_\pitchfork\times T^k$ for any $k$-dimensional torus. If $\operatorname{Sc}(g)\geq n(n-1)$, set $\ell_j=\operatorname{dist}(\partial_{j_-},\partial_{j_+})$. Gromov's square inequality. Then

$$\sum_{j=1}^m\frac{1}{\ell_j^2}\geq\frac{n^2}{4\pi^2},$$

and consequently

$$\min_{1\leq j\leq m}\operatorname{dist}(\partial_{j_-},\partial_{j_+})\leq\sqrt{m}\,\frac{2\pi}{n}.$$

The result is a conjectural statement in the relevant high-dimensional cases: the paper proves it in the spin case with a suboptimal constant, while Gromov's original minimal-surface proof in dimensions at least $9$ depends on unpublished or generalized results. It gives scalar-curvature obstructions to large distances between opposite faces of cube-like boundaries.