A priori interior estimates for special Lagrangian curvature equations

We establish a priori interior curvature estimates for the special Lagrangian curvature equations in both the critical phase and convex cases. The supercritical case, however, is distinct from the special Lagrangian equations. In dimension two, we observe that this curvature equation is equivalent to the equation arising in the optimal transportation problem with a "relative heat cost" function, as discussed in Brenier's paper. When 0 < Θ < π/2 (supercritical phase), the equation violates the Ma-Trudinger-Wang condition. However, Loeper's counterexample for general optimal transport problems does not directly apply here, as this concerns a specific optimal transport problem with fixed density functions. Moreover, the interior gradient estimates for this curvature equation are simpler than those for the special Lagrangian equations. We have demonstrated that these gradient estimates also hold for subcritical phases. It is worth noting that for the special Lagrangian equation, particularly in subcritical phases, the interior gradient estimate remains an open problem. This is joint work with Xingchen Zhou.