Special Lagrangian webbing

TL;DR

This paper constructs special Lagrangian cylinders near intersection points in Calabi-Yau manifolds, advancing the understanding of geodesics of positive Lagrangian submanifolds and their regularity.

Contribution

It introduces a method for proving $C^{1,1}$ regularity of geodesics at non-smooth loci and constructs new examples of non-invariant $C^{1,1}$ solutions in arbitrary dimensions.

Findings

Constructed families of special Lagrangian cylinders near intersection points.

Proved $C^{1,1}$ regularity of geodesics persists under small endpoint perturbations.

Provided the first examples of non-invariant $C^{1,1}$ solutions in arbitrary dimensions.

Abstract

We construct families of imaginary special Lagrangian cylinders near transverse Maslov index $0$ or $n$ intersection points of positive Lagrangian submanifolds in a general Calabi-Yau manifold. Hence, we obtain geodesics of open positive Lagrangian submanifolds near such intersection points. Moreover, this result is a first step toward the non-perturbative construction of geodesics of closed positive Lagrangian submanifolds. Also, we introduce a method for proving $C^{1,1}$ regularity of geodesics of positive Lagrangians at the non-smooth locus. This method is used to show that $C^{1,1}$ geodesics of positive Lagrangian spheres persist under small perturbations of endpoints, improving the regularity of a previous result of the authors. In particular, we obtain the first examples of $C^{1,1}$ solutions to the positive Lagrangian geodesic equation in arbitrary dimension that are not invariant under isometries. Along the way, we study geodesics of positive Lagrangian linear subspaces in a complex vector space, and prove an a priori existence result in the case of Maslov index $0$ or $n.$ Throughout the paper, the cylindrical transform introduced in previous work of the authors plays a key role.