Convexity and the degenerate special Lagrangian equation

Abstract

In 2015 Rubinstein--Solomon introduced the degenerate special Lagrangian equation (DSL) that governs geodesics in the space of positive Lagrangians, showed that subsolutions in the top branch of DSL are convex in space, and raised the question of whether they should be convex in space-time and whether subsolutions in the second branch possess any convexity properties. In 2019, Darvas--Rubinstein gave a partial answer to the first problem by showing subsolutions in the top branch must be bi-convex. We settle both questions. The key new ingredient is a space-time coordinate transformation that preserves the space-time Lagrangian angle and allows for a partial $C^2$ estimate. This also shows that the top two branches of the DSL subequation have a $\star$-product structure in the sense of Ross--Witt-Nystr\"om.