Geodesics of positive Lagrangians from special Lagrangians with boundary

TL;DR

This paper establishes a correspondence between geodesics in the space of positive Lagrangians and special Lagrangian cylinders, providing new examples of smooth solutions to the geodesic equation in arbitrary dimensions.

Contribution

It introduces the cylindrical transform linking geodesics to special Lagrangian cylinders and demonstrates the existence of smooth, non-invariant solutions in high dimensions.

Findings

Geodesics correspond to families of special Lagrangian cylinders.

The solution space of the special Lagrangian equation with boundary conditions is smooth and one-dimensional.

The space of positive Lagrangian pairs connected by a geodesic is open.

Abstract

Geodesics in the space of positive Lagrangian submanifolds are solutions of a fully non-linear degenerate elliptic PDE. We show that a geodesic segment in the space of positive Lagrangians corresponds to a one parameter family of special Lagrangian cylinders, called the cylindrical transform. The boundaries of the cylinders are contained in the positive Lagrangians at the ends of the geodesic. The special Lagrangian equation with positive Lagrangian boundary conditions is elliptic and the solution space is a smooth manifold, which is one dimensional in the case of cylinders. A geodesic can be recovered from its cylindrical transform by solving the Dirichlet problem for the Laplace operator on each cylinder. Using the cylindrical transform, we show the space of pairs of positive Lagrangian spheres connected by a geodesic is open. Thus, we obtain the first examples of strong solutions to the geodesic equation in arbitrary dimension not invariant under isometries. In fact, the solutions we obtain are smooth away from a finite set of points.