Extend Calabi’s original method to higher dimensions

Develop Calabi’s original method for proving Bernstein’s theorem for the real Monge–Ampère equation in dimensions higher than five.

Background

The paper discusses Calabi’s third-order estimates for convex solutions of the real Monge–Ampère equation and notes that Calabi’s original method establishes Bernstein-type results only in dimensions up to five. Although Pogorelov extended the theorem to all dimensions by a different method, the authors identify the adaptation of Calabi’s method to higher dimensions as an unresolved problem.

References

Recall that Calabi's theorem was in fact extended to all dimensions by Pogorelov , but Pogorelov's method is different, and it is an open problem to make Calabi's original method work in higher dimensions.

— Convex functions with symplectic Hessian  (2608.23236 - Arellano, 24 Aug 2026) in Section 1, Introduction