Cayley cones ruled by 2-planes: desingularization and implications of the twistor fibration (Q1011992)

Cayley submanifolds were introduced by \textit{R. Harvey} and \textit{H.B. Lawson} [Acta Math. 148, 47--157 (1982; Zbl 0584.53021)]. They are 4-dimensional calibrated submanifolds of 8-dimensional Riemannian manifolds with holonomy \(\text{Spin}(7)\). In the present paper the author studies 2-ruled Cayley 4-folds. These are triples \((M, \pi, \Sigma)\) where \(M^4 \subset \mathbb{O}\) is a Cayley submanifold, \(\Sigma \) is a smooth surface and \(\pi : M \rightarrow \Sigma\) is a map such that any fibre \(\pi^{-1} (\sigma)\) is an affine 2-plane in \(\mathbb{O}\). If a continuous orientation of each \(\pi^{-1}(\sigma)\) is fixed then \(M\) is called r-oriented. This yields a map \(\gamma: \Sigma \rightarrow G(2, \mathbb{O})\) in the Grassmannian of oriented 2-planes. The 2-ruled 4-fold is non-degenerate if \(\gamma\) is an immersion. The set \(M_0:= \cup_{\sigma \in \Sigma} \gamma(\sigma)\) is a cone in \(\mathbb{O}\). It is called the asymptotic cone of the 2-ruled 4-fold \((M,\pi, \Sigma)\). The Grassmannian \(G(2, \mathbb{O})\) is homogeneous under \( \text{Spin}(7) \) and admits a \(\text{Spin}(7)\)-invariant nonintegrable complex structure (see Prop. 2.1). Non-degenerate \(r\)-oriented 2-ruled Cayley cones are in correspondence with immersions \(\gamma\) which are pseudoholomorphic with respect to this structure. This result is analogous to another one by the author regarding coassociative cones (\textit{D. Fox} [Asian J. Math. 11, No.~4, 535--554 (2007; Zbl 1148.53041)]). Using the exceptional isomorphism \(\text{Spin}(6) \cong \text{SU}(4)\) one gets a ``twistor'' fibration \(\mathcal{J} : G(2, \mathbb{O}) \rightarrow S^6\) (see \S 5) which links pseudoholomorphic curves in \(G(2,\mathbb{O})\) to minimal surfaces in \(S^6\). This is reviewed in \S 7 building on earlier work of \textit{S. Salamon} [Lect. Notes Math. 1164, 161--224 (1985; Zbl 0591.53031)] and \textit{J.H. Rawnsley} [Lect. Notes Math. 1164, 85--159 (1985; Zbl 0592.58009)]. In \S 9 it is shown that any compact Riemann surface admits a pseudoholomorphic immersion in \(G(2,\mathbb{O})\) (see Thm 9.1 and Rem. 9.2). In the last section, building upon [\textit{R.L. Bryant}, in: Perspectives in Riemannian geometry. CRM Proceedings and Lecture Notes 40, 63--98 (2006; Zbl 1102.53036)], \textit{D. Joyce} [Proc. Lond. Math. Soc., III. Ser. 85, No.~1, 233--256 (2002; Zbl 1023.53034)] and \textit{J. Lotay} [J. Lond. Math. Soc., II. Ser. 74, No.~1, 219--243 (2006; Zbl 1113.53036)] the author studies the space of 2-ruled Cayley 4-folds with given asymptotic cone. Theorem 10.2 states that if \(M_0\) is a 2-ruled Cayley cone associated to a pseudoholomorphic immersion \(\gamma\) of sufficiently negative degree, then \(M_0\) is the asymptotic cone of a nonconical 2-ruled Cayley 4-fold.