Associative Submanifolds of the 7-Sphere

TL;DR

This paper classifies associative submanifolds in the 7-sphere, provides explicit examples, studies special curvature conditions, and constructs deformations, advancing understanding of minimal submanifold geometry in high-dimensional spheres.

Contribution

It offers the first explicit example of an associative 3-fold in S^7 not derived from other geometries and explores their curvature constraints and deformation families.

Findings

Explicit example of a novel associative 3-fold in S^7.

Classification of associative group orbits.

Construction of associative 3-folds with deformation families.

Abstract

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 K\"ahler 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.