Coassociative Cones Ruled by 2-planes

Abstract

Abstract. It is shown that coassociative cones in R 7 that are r-oriented and ruled by 2-planes are equivalent to CR-holomorphic curves in the oriented Grassmanian of 2-planes in R 7 . The geometry of these CR-holomorphic curves is studied and related to holomorphic curves in S 6 . This leads to an equivalence between associative cones on one side and the coassociative cones whose second fundamental form has an O(2) symmetry on the other. It also provides a number of methods for explicitly constructing coassociative 4-folds. One method leads to a family of coassociative 4-folds whose members are neither cones nor are ruled by 2-planes. This family directly generalizes the original family of examples provided by Harvey and Lawson when they introduced coassociative geometry.Key words. Differential geometry, Calibrations, Coassociative AMS subject classifications. Primary 53C381. Introduction. In 1982 Harvey and Lawson introduced coassociative 4-folds in R 7 as an example of a calibrated geometry [6]. Using the perspective of exterior differential systems they proved a local existence theorem. Harvey and Lawson offered a single finite dimensional family of explicit examples (see section 9). In 1985 Mashimo classified the coassociative cones whose links are homogeneous [15]. Then for almost twenty years the literature was quiet on the subject. The silence was broken in 2004 when techniques used for studying special Lagrangian geometry (which is another type of calibrated geometry) began to be applied to coassociative geometry [13], [7], [11]. This renewal of activity was motivated by Joyce's construction of compact manifolds with G 2 holonomy [9] and the appearance of coassociative geometry in M-theory [1].In [13] Lotay studied coassociative 4-folds in R 7 that are ruled by 2-planes. He gave a local existence result and presented a method for using a holomorphic vector field on a naturally associated Riemann surface to deform a coassociative cone that is ruled by 2-planes. His work is an extension of Joyce's work on ruled special Lagrangian 3-folds [10].This article revisits the geometry of coassociative cones that are ruled by 2-planes using the methods introduced by Bryant in [3]. This perspective realizes the Riemann surface that appeared in [13] as a CR-holomorphic curve for a G 2 -invariant CR-structure onG(2, 7), the oriented Grassmanian of 2-planes in R 7 . In fact, Proposition 7.2 asserts that r-oriented 2-ruled coassociative cones are equivalent to CRholomorphic curves inG(2, 7).Studying the geometry of CR-holomorphic curves leads to new methods for constructing coassociative cones that are ruled by 2-planes. These methods arise from the close relationship between CR-holomorphic curves inG(2, 7) and holomorphic curves 1 in S 6 . This relationship can be interpreted as an equivalence between the associative cones and a certain family of coassociative cones. The null-torsion holomorphic curves in S 6 also form the backbone for a new family of coassociative 4-folds that directly generalizes the family introduc...

Ruled Lagrangian submanifolds of the 6-sphere

“…Remarks This is the natural analogue of the characterisation of ruled special Lagrangians in C 3 given in [3,Theorem 6]. We should stress that our proposition is little more than a repackaging of the material given in [12,Proposition 7.2].…”

“…Lagrangians satisfying certain curvature conditions are classified in [6] and [8]. Ruled Lagrangian submanifolds in S 6 are equivalent to coassociative cones in R 7 ruled by 2-planes, which are studied in [12] and by the author in [20]. A special family of ruled Lagrangians is analysed in [25].…”

“…We now make an elementary observation. By [12,Proposition 7.2], there is a correspondence between 2-ruled coassociative cones in Im O and certain surfaces in the Grassmannian of oriented 2-planes in Im O, Gr + (2, Im O). Notice that Gr + (2, Im O) is naturally isomorphic to the space C of oriented geodesic circles in S 6 : simply identify an oriented 2-plane in Im O with its intersection with S 6 .…”

Cayley cones ruled by $2$-planes: desingularization and implications of the twistor fibration

“…This almost complex structure is referred to as J 2 in [25] and the nonintegrability is given in Proposition 3.2 of [25]. The rest of the proof is similar to that of the analogous result for 2-ruled coassociative cones, Proposition 7.2 of [14], so here we only sketch the proof of the second statement. The Cayley conditions reduce to the vanishing of certain 2-forms on the surface γ(Σ) ⊂ G(2, O) (Equation (2.7)) and these 2-forms are the real and imaginary parts of (2, 0)-forms for the almost complex structure on G(2, O) that is defined by declaring the ζ i of Equation (2.8) to be of type (1,0).…”

Desingularization of coassociative 4-folds with conical singularities: Obstructions and applications

“…Moreover, C(Γ) admits a dilation family of smoothings A(Γ) which converge with rate − 3 2 to (possibly a finite cover of) C(Γ) (c.f. [3,Theorem 9.3]). If C = C(Γ) and A = A(Γ) is AC, then our theory applies whenever the matching condition holds, which is a non-trivial constraint.…”