Scalar Curvature Question [?60]: Also it is unclear if there are (non-trivial) inequalities of this kind for other exotic spheres

Definitions and mathematical statement

An exotic $n$-sphere is a smooth closed manifold $\Sigma^n$ homeomorphic to $S^n$ but not diffeomorphic to the standard smooth sphere. The dimension must be one in which the intended exotic spheres exist; no dimension range is specified in this record.

The complete mathematical request in the stored statement is whether there are nontrivial inequalities of this kind for other exotic spheres. The antecedent inequality, its geometric variables, hypotheses, constants, comparison object, and the previously discussed exotic spheres are absent. Consequently no particular scalar-curvature, immersion, diameter, or rigidity inequality can be extracted without inventing its content. The words this kind and other are essential unresolved cross-references, not authorizations to choose an arbitrary inequality.

The supplied research summary places the question in a scalar-curvature/rigidity context and says that many exotic spheres in some dimensions admit positive-scalar-curvature metrics via surgery theory under appropriate obstruction conditions. This does not assert that every exotic sphere admits such a metric, or that all have vanishing spin-index obstruction. It supplies no explicit inequality or verified result distinguishing the spheres intended by the fragment. The draft therefore preserves an unresolved existence-of-inequalities programme rather than presenting an invented quantified bound.

Short English statement

Are there nontrivial geometric inequalities for the other exotic spheres discussed by the source? The record omits both the antecedent inequality and the spheres being compared, so the mathematical target cannot be completed from this fragment.

Sources

• Local UnsolvedMath registry, permanent ID $6700056$, exact source code catalogue entry; printed scalar-curvature tag [?60], occurrence 59. Proposer Misha Gromov, 2017.
• Gromov, 101 Questions, Problems and Conjectures around Scalar Curvature (2017), recorded PDF page 59; https://www.ihes.fr/~gromov/wp-content/uploads/2018/08/101-problemsOct1-2017.pdf. Recorded access 2026-07-29; pdf-text-normalized extraction.
• Supplied OPEN-TRIAGE research summary mentions Gromov--Lawson surgery and spin-index obstructions as background, but gives no precise publication citation, dimension/obstruction theorem, or inequality matching this fragment.