A minimum principle for Lagrangian graphs

Abstract: The classical minimum principle is foundational in convex and complex analysis and plays an important role in the study of the real and complex Monge-Ampere equations. This note establishes a minimum principle in Lagrangian geometry. This principle relates the classical Lagrangian angle of Harvey-Lawson and the space-time Lagrangian angle introduced recently by Rubinstein-Solomon. As an application, this gives a new formula for solutions of the degenerate special Lagrangian equation in space-time in terms of the (time) partial Legendre transform of a family of solutions of obstacle problems for the (space) non-degenerate special Lagrangian equation.