Associative submanifolds of the 7-sphere

Associative submanifolds of the 7-sphere S-7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R-8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C-4, as well as Lagrangian submanifolds of the nearly Kahler 6-sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3-fold in S-7 which does not arise from other geometries. We then study associative 3-folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2-planes in R-8 and isotropic minimal surfaces in S-6. We also prove that associative 3-folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3-folds satisfying Chen's equality which have an S-1-family of global isometric deformations using harmonic 2-spheres in S-6.