Scalar Curvature Question [?71]: [a] Are the diameters diamc(Lip1(Bn(R)→S))) bounded for a large fixedc and R →∞if Hi(S, R) = 0 for i = 1, 2,

Definitions and mathematical statement

Let $B^n(R)$ be the Euclidean radius-$R$ ball and let $S$ be the intended metric target. Define \[ \operatorname{Lip}_1(B^n(R),S)=\{f:B^n(R)\to S:d_S(f(x),f(y))\leq|x-y|\text{ for all }x,y\}. \] Assume $H_i(S;\mathbb R)=0$ for $i=1,\ldots,n$, with real coefficients. The source asks whether \[ D_c(R)=\operatorname{diam}_c\left(\operatorname{Lip}_1(B^n(R),S)\right) \] stays bounded as $R\to\infty$ for a sufficiently large fixed $c$. A basic uniform interpretation is that suitable finite $c,C,R_0$, depending on the fixed target and dimension but not on $R$, satisfy $D_c(R)\leq C$ for all $R\geq R_0$. The parameter $c$ must not grow with $R$.

The subscripted diameter is not defined in the supplied record. One possible mapping-space length convention would use the uniform distance $d_\infty(f,g)=\sup_x d_S(f(x),g(x))$, allow homotopies through $c$-Lipschitz maps, and minimize their $d_\infty$-length between endpoint maps. This is explanatory only: the source may use a different path norm, and it does not say whether disconnected pairs have infinite distance or diameters are taken componentwise. The geometry and connectedness of $S$, ball-boundary conventions, and the exact quantifier on sufficiently large $c$ are also absent. Thus the boundedness target is retained without inventing the missing diameter definition.

Short English statement

If the target has vanishing real homology through degree $n$, do the source's controlled mapping-space diameters of $1$-Lipschitz maps from larger and larger $n$-balls stay uniformly bounded with fixed control $c$? The controlled distance is not defined in the extract.

Sources

• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?71], occurrence 70, PDF page 72, subpart [a], https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; accessed 2026-07-29, PDF-text-normalized extraction. The provenance describes the public author PDF as incomplete.
• The supplied summary associates this with Gromov's mapping-space path-length ideas and refers to "Length of paths" without a complete publication identifier or the required distance definition.
• Local registry data/unsolvedmath/problems.json, permanent UnsolvedMath ID $6700067$, source code catalogue entry, Scalar Curvature Question [?71]. Dataset attribution: Ulam AI and the UnsolvedMath Contributors.