2-Ruled Calibrated 4-folds in R^7 and R^8
  In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the author in math.DG/0401123 on ruled associative 3-folds in R^7. We say that a 2-ruled 4-fold M is r-framed if we are given an oriented basis for each fibre in a smooth manner, and in such circumstances we may write M in terms of orthogonal smooth maps phi_1,phi_2:Sigma-->S^(n-1) and a smooth map psi:Sigma-->R^n. We focus on 2-ruled Cayley 4-folds since coassociative and special Lagrangian 4-folds can be considered as special cases.   The main result is on non-planar, r-framed, 2-ruled Cayley 4-folds in R^8, which characterises the Cayley condition in terms of 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 deduce that, for a fixed non-planar, r-framed, 2-ruled Cayley cone M_0, the space of r-framed 2-ruled Cayley 4-folds M which have asymptotic cone M_0 has the structure of a vector space.   We give a means of constructing 2-ruled Cayley 4-folds M starting from a 2-ruled Cayley cone M_0, satisfying a certain condition, using holomorphic vector fields such that M_0 is the asymptotic cone of M. We use this to construct explicit examples of U(1)-invariant 2-ruled Cayley 4-folds asymptotic to a U(1)^3-invariant 2-ruled Cayley cone. Examples are also given based on ruled calibrated 3-folds in C^3 and R^7 and complex cones in C^4. 
Jason Lotay