Special Lagrangian cones

We study special Lagrangian cones in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /] with isolated singularities especially the case n = 3. Our main result constructs an infinite family of special Lagrangian cones in [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="02i" /] 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 [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="03i" /] 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.