Scalar Curvature Question [?24]: (i) It is unclear if the last step in the above argument is truly needed: conceivably, maps Φ ∶Sn−1→U(N) with

Definitions and mathematical statement

Let $U(N)=\{A\in M_N({\mathbb{C}}):A^*A=I\}$. Once metrics $d_S$ on $S^{n-1}$ and $d_U$ on $U(N)$ are fixed, define \[ \operatorname{Lip}(\Phi)=\sup_{x\ne y}\frac{d_U(\Phi(x),\Phi(y))}{d_S(x,y)},\qquad \mathcal L_{n,N}=\{\Phi:S^{n-1}\to U(N):\operatorname{Lip}(\Phi)<1/2\}. \] The threshold $1/2$ is the likely fraction reading of the extracted "1 2" and requires confirmation. Gromov suggests that these maps may be deformable to constant maps continuously in $\Phi$, for all $n$. A parameter-dependent null-homotopy formulation would ask for a continuous map \[ H:\mathcal L_{n,N}\times[0,1]\longrightarrow C(S^{n-1},U(N)) \] with $H(\Phi,0)=\Phi$ and $H(\Phi,1)$ constant on $S^{n-1}$ for every $\Phi$. The topology on the map spaces must be specified; the source does not state it. It also does not say whether $H(\Phi,t)$ must remain in $\mathcal L_{n,N}$, satisfy some other Lipschitz bound, or fix constant maps throughout.

This asks for a continuously chosen deformation, not merely a separate null-homotopy for each map. Nor does deforming to the space of constant maps assert contractibility to one fixed constant map. The preceding argument and its "last step", which motivates the suggestion, are not supplied. The metrics, the range of $n$, and whether the assertion is uniform in $N$ are likewise unstated.

Short English statement

Can sufficiently small Lipschitz maps from spheres to unitary groups be deformed to constant maps by a choice that varies continuously with the original map, in every dimension? Metric normalizations and the required topology and Lipschitz control of the deformation are missing.

Sources

• Local UnsolvedMath registry, permanent ID 6700025, source code catalogue entry; Misha Gromov, 2017; update 2026-08-06.
• Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?24], occurrence 25, PDF page 36, passage (i): https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Registry access date 2026-07-29; normalized PDF extraction; incomplete public author-hosted PDF; status and reuse terms require review.
• The supplied summary associates the question with a K-area/index argument and Lipschitz-topology contractibility but gives no additional reference or precise formulation. Its description as involving high Lipschitz constants does not match the extracted small-threshold inequality.