Resolve equality in Calabi’s differential inequality for dimensions at least three

Determine whether equality in Calabi’s differential inequality \(\Delta R^{\alpha}\ge 2\alpha\frac{n+1}{n(n-1)}R^{\alpha+1}\) can occur at a single point where the scalar curvature is positive for Hessian metrics associated with convex solutions of the real Monge–Ampère equation in dimensions \(n\ge 3\).

Background

Calabi had claimed that equality in the differential inequality could not occur even at a single point with positive scalar curvature. The paper gives a counterexample to that stronger claim in dimension two, but leaves the corresponding question unresolved in dimensions three and higher.

References

In fact, Example \ref{n=2} is an explicit counterexample to this stronger claim when $n=2$. The case $n 3$ remains open.

— Convex functions with symplectic Hessian  (2608.23236 - Arellano, 24 Aug 2026) in Remark following Proposition \ref{prop:DiffIneqR}, Section 3.2, “Comparison with some results of Calabi”