A generalization of Cayley submanifolds

TL;DR

This paper studies Cayley submanifolds in 4-dimensional Kaehler manifolds, characterizing their properties and minimality conditions in Calabi-Yau and Kaehler-Einstein settings.

Contribution

It generalizes the concept of Cayley submanifolds and classifies minimal Cayley submanifolds in specific Kaehler ambient manifolds.

Findings

Minimal Cayley submanifolds in Calabi-Yau are as defined by Harvey and Lawson.

In Kaehler-Einstein manifolds with non-zero scalar curvature, minimal Cayley submanifolds are either complex or Lagrangian.

The paper establishes basic properties of Cayley submanifolds in complex dimension 4.

Abstract

Given a Kaehler manifold of complex dimension 4, we consider submanifolds of (real) dimension 4, whose Kaehler angles coincide. We call these submanifolds Cayley. We investigate some of their basic properties, and prove that (a) if the ambient manifold is a Calabi-Yau, the minimal Cayley submanifolds are just the Cayley submanifolds as defined by Harvey and Lawson; (b) if the ambient is a Kaehler-Einstein manifold of non-zero scalar curvature, then minimal Cayley submanifolds have to be either complex or Lagrangian.