Special Lagrangians with Cylindrical Tangent Cones

TL;DR

This paper constructs new examples of special Lagrangian submanifolds with isolated singularities and cylindrical tangent cones, demonstrating novel geometric properties in higher-dimensional complex spaces.

Contribution

It introduces explicit constructions of special Lagrangian submanifolds with cylindrical tangent cones and unique connectivity properties near singularities.

Findings

Existence of special Lagrangian submanifolds with cylindrical tangent cones.

Examples where the submanifold minus the singularity is connected.

The tangent cone can be a pair of transverse planes.

Abstract

We construct new examples of special Lagrangian submanifolds $Y\subset \mathbf{C}^{n+1}$, $n\geq 3$ in a neighborhood of the origin, with an isolated singularity, but with cylindrical tangent cone $C\times\mathbf{R}$. Moreover, $Y\setminus\{0\}$ is connected while $(C\setminus\{0\})\times\mathbf{R}$ is not. Such examples exist, for example, when $C$ is a pair of transverse planes.