Special Lagrangian Cones

TL;DR

This paper constructs and analyzes special Lagrangian cones in complex three-space, revealing new examples with toroidal links, proving regularity results, and developing asymptotically conical submanifolds.

Contribution

It introduces an infinite family of special Lagrangian cones with toroidal links and establishes a regularity theorem for cones with spherical links in ^3.

Findings

Constructed an infinite family of cones with toroidal links.

Proved that cones with spherical links in ^3 are planes.

Developed a family of asymptotically conical special Lagrangian submanifolds.

Abstract

We study special Lagrangian cones in $\C^n$ with isolated singularities. Our main result constructs an infinite family of special Lagrangian cones in $\C^3$ each of which has a toroidal link. We obtain a detailed geometric description of these tori. We prove a regularity result for special Lagrangian cones in $\C^3$ with a spherical link -- any such cone must be a plane. We also construct a one-parameter family of asymptotically conical special Lagrangian submanifolds from any special Lagrangian cone.