Convexity and the degenerate special Lagrangian equation

TL;DR

This paper investigates convexity properties of solutions to the degenerate special Lagrangian equation, establishing space-time convexity and bi-convexity in certain branches using a novel coordinate transformation.

Contribution

It proves that subsolutions in the top branch are convex in space-time and that top branches are bi-convex, resolving open questions in the theory.

Findings

Subsolutions in the top branch are convex in space-time.

Top two branches of the DSL have a $igstar$-product structure.

A space-time coordinate transformation preserves the Lagrangian angle.

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.