Scalar Curvature Question [?54]: ExtremalityofConcaveSphericalBalls

Definitions and mathematical statement

Let $(S^n,g_{\mathrm{sph}})$ be the unit round sphere, so $\operatorname{Sc}_{g_{\mathrm{sph}}}=n(n-1)$, and let $B(R)$ be a geodesic ball of radius $R\ge\pi/2$. The source calls these concave spherical balls. The open-versus-closed ball convention and treatment of $R=\pi$ or larger are not stated; in the closed-ball convention $R\ge\pi$ gives the whole sphere.

The intended length comparison is usually the quadratic-form inequality \[ g_x(v,v)\ge(g_{\mathrm{sph}})_x(v,v) \qquad(x\in B(R),\ v\in T_xB(R)), \] although the source's word "greater" does not distinguish strict from non-strict comparison. Its length-extremality conjecture says that no such Riemannian metric can have \[ \operatorname{Sc}_g(x)>n(n-1)\qquad\text{at every }x\in B(R). \] Thus, under the non-strict metric comparison reading, the target is \[ \forall g\ge g_{\mathrm{sph}}|_{B(R)}\quad \exists x\in B(R):\operatorname{Sc}_g(x)\le n(n-1). \] The statement forbids strict scalar-curvature improvement everywhere; it does not assert a pointwise upper bound everywhere or an isometry conclusion in the equality case.

No agreement of metrics at the boundary, mean-curvature bound, prescribed boundary geometry, or extension across the boundary is supplied. The ball convention, smoothness up to any included boundary, dimension range, and metric-comparison convention must be fixed. These omissions are not filled by borrowing boundary hypotheses from a separate hemisphere rigidity theorem.

Short English statement

Can a metric that length-dominates the round metric on a spherical ball of radius at least $\pi/2$ have scalar curvature strictly above the spherical value everywhere? The conjecture says no, with no supplied additional boundary condition.

The research summary reports sphere and some hemisphere/ball results but gives no precise hypotheses, and says the general radius/dimension case remains open. It names Llarull, "Brendle--Marques--Nevens," and "Schoen--Yau/Huisken" without exact references. Those informal rigidity attributions are not treated as verification of this boundary-unconstrained extremality statement. The partial-progress label and 2017 passage are retained with current status needing review.

Sources

• Local UnsolvedMath registry, permanent ID 6700051, source code catalogue entry, scalar-curvature Question [?54], "Extremality of Concave Spherical Balls," proposed by Misha Gromov, 2017. Dataset curation and metadata: Ulam AI and UnsolvedMath Contributors, CC BY 4.0.
• Misha Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), printed tag [?54], occurrence 53, PDF page 57, https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf; normalized pdf-text extraction, accessed 2026-07-29. The extracted final reference marker is incomplete.
• The summary's Llarull, "Brendle--Marques--Nevens," and "Schoen--Yau/Huisken" attributions have no full entries or exact hypotheses. Spellings and asserted roles are not independently verified or corrected.
• Source-specific provenance: the author-hosted PDF is incomplete and publicly available; current status and statement-reuse terms NEEDS_REVIEW, separately from the dataset licence.