2-RULED CALIBRATED 4-FOLDS IN ${\mathbb{R}}^7$ AND ${\mathbb{R}}^8$

Abstract

This article introduces the notion of 2-ruled 4-folds: submanifolds of ${\mathbb{R}}^n$ fibred over a 2-fold $\Sigma$ by affine 2-planes. This is motivated by a paper by Joyce and previous work of the present author. A 2-ruled 4-fold $M$ is r-framed if an oriented basis is smoothly assigned to each fibre, and then we may write $M$ in terms of orthogonal smooth maps $\phi_1,\phi_2:\Sigma\rightarrow\mathcal{S}^{n-1}$ and a smooth map $\psi:\Sigma\rightarrow{\mathbb{R}}^n$. We focus on 2-ruled Cayley 4-folds in ${\mathbb{R}}^8$ as certain other calibrated 4-folds in ${\mathbb{R}}^7$ and ${\mathbb{R}}^8$ can be considered as special cases. The main result characterizes non-planar, r-framed, 2-ruled Cayley 4-folds, using a coupled system of nonlinear, first-order, partial differential equations that $\phi_1$ and $\phi_2$ satisfy, and another such equation on $\psi$ which is linear in $\psi$. We give a means of constructing 2-ruled Cayley 4-folds starting from particular 2-ruled Cayley cones using holomorphic vector fields. This is used to give explicit examples of ${\mathbin{\rm U}}(1)$-invariant 2-ruled Cayley 4-folds asymptotic to a ${\mathbin{\rm U}}(1)^3$-invariant 2-ruled Cayley cone. Examples are also given based on ruled calibrated 3-folds in ${\mathbb{C}}^3$ and ${\mathbb{R}}^7$ and complex cones in ${\mathbb{C}}^4$.