The degenerate special Lagrangian equation

Abstract

This article introduces the degenerate special Lagrangian equation (DSL) and develops the basic analytic tools to construct and study its solutions. The DSL governs geodesics in the space of positive graph Lagrangians in ${\mathbb{C}}^n.$ Existence of geodesics in the space of positive Lagrangians is an important step in a program for proving existence and uniqueness of special Lagrangians. Moreover, it would imply certain cases of the strong Arnold conjecture from Hamiltonian dynamics. We show the DSL is degenerate elliptic. We introduce a space-time Lagrangian angle for one-parameter families of graph Lagrangians, and construct its regularized lift. The superlevel sets of the regularized lift define subequations for the DSL in the sense of Harvey--Lawson. We extend the existence theory of Harvey--Lawson for subequations to the setting of domains with corners, and thus obtain solutions to the Dirichlet problem for the DSL in all branches. Moreover, we introduce the calibration measure, which plays a rôle similar to that of the Monge--Ampère measure in convex and complex geometry. The existence of this measure and regularity estimates allow us to prove that the solutions we obtain in the outer branches of the DSL have a well-defined length in the space of positive Lagrangians.

42 pages; implemented referee suggestions, added appendix B on geometric formulation of DSL, modified Theorems 1.2 and 8.1 to reflect correction in Lemma 8.4

References in corpus (3)

• Smooth and singular Kahler-Einstein metrics
• A Floer fundamental group
• Geodesics of positive Lagrangians in Milnor fibers

Cited by in corpus (6)

• (1,1) forms with specified Lagrangian phase: A priori estimates and algebraic obstructions
• The space of almost calibrated $(1,1)$ forms on a compact Kähler manifold
• Geodesics of positive Lagrangians in Milnor fibers
• Geodesics of positive Lagrangians from special Lagrangians with boundary
• A priori estimates for a generalised Monge-Ampère PDE on some compact Kähler manifolds
• On the existence of geodesic connecting Lagrangian graphs in ${\mathbb{C}}^n$