Scalar Curvature Question [?68]: Let $\widetilde X$ be the universal cover of a Riemannian $n$-manifold $X$ homeomorphic to the $n$-torus

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a Riemannian domain, mean curvature $H$ of a smooth boundary face is the trace of its second fundamental form with the convention that the boundary of a Euclidean ball has positive mean curvature. Mean convexity means $H\geq0$. Dihedral angles are interior angles between adjacent faces, measured between their tangent spaces along their common codimension-two faces. Let $(X,g)$ be a Riemannian $n$-manifold homeomorphic to the torus $T^n=(S^1)^n$, and let $\widetilde X$ be its universal cover with the lifted metric. An exhaustion by domains is a sequence of compact domains $P_i\subset\widetilde X$ whose interiors exhaust $\widetilde X$ and with $P_i\subset P_{i+1}$. An overcubic domain has a corner structure and a degree-one map $f_i:P_i\to[0,1]^n$ respecting faces in the corner-proper sense: for every codimension-one cube face $\bar F_j$, its corresponding domain face is exactly $F_j=f_i^{-1}(\bar F_j)$. Consequently each lower-dimensional face is sent to its corresponding cube face. Degree is the integer by which the induced map multiplies the oriented relative fundamental class in top-dimensional homology. The phrase "non-positive scalar curvature at infinity" in this question refers to the proposed exhaustion below, without a pointwise sign assumption on the scalar curvature of $g$.

Statement recorded in the supplied source.

Prove that $\widetilde X$ admits an exhaustion by overcubic domains $P_i$ with corners such that every codimension-one face has strictly positive mean curvature and every dihedral angle is at most $\pi/2$. [S2]

Short English statement

Can the universal cover of any Riemannian torus be exhausted by domains mapping with degree one to a cube, with positive face mean curvature and no dihedral angle larger than a right angle?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6700064 (AMR-066-0064). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Misha Gromov, Four Lectures on Scalar Curvature, arXiv:1908.10612v6 (9 July 2021), §3.13, Exercise and full hint, printed p. 168; corner-map definitions in §3.18, printed p. 186. https://arxiv.org/abs/1908.10612v6.