The degenerate special Lagrangian equation on Riemannian manifolds

TL;DR

This paper extends the degenerate special Lagrangian equation to Riemannian manifolds, establishing its role in governing geodesics in the space of positive Lagrangians and developing new analytic techniques for solutions.

Contribution

It introduces a global formulation of the Riemannian degenerate special Lagrangian equation and adapts Dirichlet duality methods to obtain continuous solutions.

Findings

The Riemannian DSL governs geodesics in positive Lagrangian spaces.

Continuous solutions to the Dirichlet problem for Riemannian DSL are obtained.

The approach generalizes Euclidean results to Riemannian manifolds.

Abstract

We show that the degenerate special Lagrangian equation, recently introduced by Rubinstein-Solomon, induces a global equation on every Riemannian manifold, and that for certain associated geometries this equation governs, as it does in the Euclidean setting, geodesics in the space of positive Lagrangians. For example, geodesics in the space of positive Lagrangian sections of a smooth Calabi-Yau torus fibration are governed by the Riemannian DSL on the base manifold. We then develop their analytic techniques, specifically modifications of the Dirichlet duality theory of Harvey-Lawson, in the Riemannian setting to obtain continuous solutions to the Dirichlet problem for the Riemannian DSL and hence continuous geodesics in the space of positive Lagrangians.