A constant rank theorem for special Lagrangian equations

We derive constant rank theorems for saddle solutions to special Lagrangian equations and the quadratic Hessian equation. The constant rank theorem asserts that, for solutions of the equations which verify a certain convexity condition, a strong minimum principle holds for the minimum eigenvalue of the Hessian. We discuss connections between this result and the rigidity and regularity of solutions, and also mention a conceptually simple approach to proving a general constant rank theorem for fully nonlinear inverse-convex elliptic equations. This is joint work with Yu Yuan.