Constructing associative 3-folds by evolution (Q871728)

The paper under review gives two methods for constructing associative 3-folds in \(\mathbb R^7\), based around the fundamental idea of evolution equations, and uses them to produce examples. It is a generalisation of the works by \textit{D. D. Joyce} [Duke Math. J. 115, No. 1, 1--51 (2002; Zbl 1023.53033)], etc., on special Lagrangian (SL) \(3\)-folds in \(\mathbb C^3\). The methods described involve the use of an affine evolution equation with affine evolution data and the area of ruled submanifolds. The author derives an affine evolution equation using affine evolution data. This is used on an example of such data to construct a 14-dimensional family of associative 3-folds. One of the main results of the paper is an explicit solution of the system of differential equations generated in a particular case to give a 12-dimensional family of associative 3-folds. Moreover, he finds a straightforward condition which ensures that the associative 3-folds constructed are closed and diffeomorphic to \(S^1\times\mathbb R^2\), rather than \(\mathbb R^3\). The final section deals with ruled associative 3-folds.