Special Lagrangian webbing

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

regularity of geodesics of positive Lagrangians at the non-smooth locus. This method is used to show that

C 1 , 1 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 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.