Higher-dimensional regularity for strictly 2-convex sigma-2 solutions

Establish interior Hessian bounds and corresponding regularity for strictly 2-convex solutions of the equation \(\sigma_2(D^2u)=\psi\) in dimensions \(n\geq 5\).

Background

The paper surveys known interior regularity results for the σ2\sigma_2 Hessian equation. Such estimates are available in dimensions three and four for strictly 2-convex solutions, while higher-dimensional cases present substantial difficulties and lack the special Lagrangian or Jacobi-inequality structures used in lower dimensions.

The unresolved issue concerns whether strictly 2-convex solutions in dimensions five and higher possess the desired interior regularity, rather than the stronger strictly convex setting for which estimates are known.

References

For higher dimensions n \geq 5, the regularity question remains open for strictly $2$-convex solutions and we refer the reader to a discussion by Mooney .

— A note on interior curvature estimates for strictly convex solutions to the equation of prescribed curvature quotient  (2608.20188 - Wang, 20 Aug 2026) in Section 1, Literature review

This leaves a longstanding open problem: For $n\ge 3$, whether a priori interior $C2$ estimates and regularity can be established for the $\sigma_2$-Hessian equation $\sigma_2(D2u)=1$ and the $\sigma_2$-curvature equation $\sigma_2(\kappa)=1$?

— Interior estimates and regularity for the scalar curvature equation in dimension 4  (2608.22748 - Fan et al., 24 Aug 2026) in Problem \ref{PROB}, Section 1, Introduction