Deformation Theory of Asymptotically Conical Coassociative 4-folds

We study coassociative 4-folds N in R^7 which are asymptotically conical to a cone C with rate lambda<1. If lambda is in the interval [-2,1) and generic, we show that the moduli space of coassociative deformations of N which are also asymptotically conical to C with rate lambda is a smooth manifold, and we calculate its dimension. If lambda<-2 and generic, we show that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds, and we give a lower bound for its expected dimension. We also derive a test for when N will be planar if lambda<-2 and we discuss examples of asymptotically conical coassociative 4-folds.Comment: 50 pages, LaTeX; v2: numerous presentation improvements and changes, some general theory of elliptic operators between weighted Banach spaces added to aid the reader; v3: further results included and proofs streamline

Constructing Associative 3-folds by Evolution Equations

This paper gives two methods for constructing associative 3-folds in R^7, based around the fundamental idea of evolution equations, and uses these methods to construct examples of these geometric objects. The paper is a generalisation of the work by Joyce in math.DG/0008021, math.DG/0008155, math.DG/0010036 and math.DG/0012060 on special Lagrangian 3-folds in C^3. The two methods described involve the use of an affine evolution equation with affine evolution data and the area of ruled submanifolds. We first give a derivation of an evolution equation for associative 3-folds from which we derive an affine evolution equation using affine evolution data. We then use this on an example of such data to construct a 14-dimensional family of associative 3-folds. One of the main result of the paper is then an explicit solution of the system of differential equations generated in a particular case to give a 12-dimensional family of associative 3-folds. We also find that there is a straightforward condition that ensures that the associative 3-folds constructed are closed and diffeomorphic to S^1xR^2, rather than R^3. In the final section we define ruled associative 3-folds and derive an evolution equation for them. This then allows us to characterise a family of ruled associative 3-folds using two real analytic maps that must satisfy two partial differential equations. We finish by giving a means of constructing ruled associative 3-folds M from r-oriented two-sided associative cones M_0 such that M is asymptotically conical to M_0 with order O(r^{-1}).Comment: 43 pages, LaTeX; minor corrections, mainly typos, and slight changes in presentation, including added reference

Associative Submanifolds of the 7-Sphere

Associative submanifolds of the 7-sphere S^7 are 3-dimensional minimal submanifolds which are the links of calibrated 4-dimensional cones in R^8 called Cayley cones. Examples of associative 3-folds are thus given by the links of complex and special Lagrangian cones in C^4, as well as Lagrangian submanifolds of the nearly K\"ahler 6-sphere. By classifying the associative group orbits, we exhibit the first known explicit example of an associative 3-fold in S^7 which does not arise from other geometries. We then study associative 3-folds satisfying the curvature constraint known as Chen's equality, which is equivalent to a natural pointwise condition on the second fundamental form, and describe them using a new family of pseudoholomorphic curves in the Grassmannian of 2-planes in R^8 and isotropic minimal surfaces in S^6. We also prove that associative 3-folds which are ruled by geodesic circles, like minimal surfaces in space forms, admit families of local isometric deformations. Finally, we construct associative 3-folds satisfying Chen's equality which have an S^1-family of global isometric deformations using harmonic 2-spheres in S^6.Comment: 42 pages, v2: minor corrections, streamlined and improved exposition, published version; Proceedings of the London Mathematical Society, Advance Access published 17 June 201

Asymptotically Conical Associative 3-folds

Given an associative 3-fold in R^7 which is asymptotically conical with generic rate less than 1, we show that its moduli space of deformations is locally homeomorphic to the kernel of a smooth map between smooth manifolds. Moreover, the virtual dimension of the moduli space is computed and shown to be non-negative for rates greater than -1, whereas the associative 3-fold is expected to be isolated for rates less than or equal to -1.Comment: 33 pages, v2: major changes for published version, mainly regarding the twisted Dirac and d-bar operator