This paper is dedicated to the study of deformations of coassociative 4-folds in a G_2 manifold which have conical singularities. We stratify the types of deformations allowed into three problems. The main result for each problem states that the moduli space is locally homeomorphic to the kernel of a smooth map between smooth manifolds. In each case, the map in question can be considered as a projection from the infinitesimal deformation space onto the obstruction space. Thus, when there are no obstructions the moduli space is a smooth manifold. Furthermore, we calculate a lower bound on the expected dimension of the moduli space. Finally we show that, in weakening the condition on the G_2 structure of the ambient 7-manifold, there is a generic smoothness result for the moduli spaces of deformations corresponding to our second and third problems.
We study the problem of desingularizing coassociative conical singularities via gluing, allowing for topological and analytic obstructions, and discuss applications. This extends the author's earlier work on the unobstructed case. We interpret the analytic obstructions geometrically via the obstruction theory for deformations of conically singular coassociative 4-folds, and thus relate them to the stability of the singularities. We use our results to describe the relationship between moduli spaces of coassociative 4-folds with conical singularities and those of their desingularizations. We als…
Given a coassociative 4-fold N with a conical singularity in a varphi-closed 7-manifold M (a manifold endowed with a distinguished closed 3-form varphi), we construct a smooth family, {N'(t): t\in(0,tau)} for some tau>0, of (smooth, nonsingular,) compact coassociative 4-folds in M which converge to N in the sense of currents, in geometric measure theory, as t tends to 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4-fold in R^7, dilated by t, then deforming the resulting compact submanifold of M to the required coassociative 4-fo…
We study the stability of coassociative 4-folds with conical singularities under perturbations of the ambient G_2 structure by defining an integer invariant of a coassociative cone which we call the stability index. The stability index of a coassociative cone is determined by the spectrum of the curl operator acting on its link. We explicitly calculate the stability index for cones on group orbits. We also describe the stability index for cones fibered by 2-planes over algebraic curves using the degree and genus of the curve and the spectrum of the Laplacian on the link. Finally we apply our r…
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 e…
Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_{2}$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in $Λ^{2}_{-} S^{4}$, which is the complete $G_{2}$-manifold constructed by Bryant and Salamon. Classifying the Lie groups which have 3- or 4-dimensional orbits, we show that the only homogeneous coassociative submanifold is the zero section of $Λ^{2}_{-} S^{4}$ up to the automorphisms and construct many cohomogeneity one examples explicitly. In particular, we obtain examples of non-compact coassociative submanifolds with conical…
Coassociative 4-folds are a particular class of 4-dimensional submanifolds which are defined in a 7-dimensional manifold M with a G_2 structure given by a `positive' differential 3-form, sometimes called G_2-form. Assuming that a G_2-form on M is closed, we study deformations of a compact coassociative submanifold N with boundary contained in fixed, codimension 1 submanifold S of M with a compatible Hermitian symplectic structure. We show that `small' coassociative deformations of N with special Lagrangian boundary in S are unobstructed and form a smooth moduli space of finite dimension not gr…
We derive formulas for the mean curvature of associative 3-folds, coassociative 4-folds, and Cayley 4-folds in the general case where the ambient space has intrinsic torsion. Consequently, we are able to characterize those G2-structures (resp., Spin(7)-structures) for which every associative 3-fold (resp. coassociative 4-fold, Cayley 4-fold) is a minimal submanifold. In the process, we obtain new obstructions to the local existence of coassociative 4-folds in G2-structures with torsion.
Bryant-Salamon constructed three 1-parameter families of complete manifolds with holonomy $\mathrm{G}_2$ which are asymptotically conical to a holonomy $\mathrm{G}_2$ cone. For each of these families, including their asymptotic cone, we construct a fibration by asymptotically conical and conically singular coassociative 4-folds. We show that these fibrations are natural generalizations of the following three well-known coassociative fibrations on $\mathbb R^7$: the trivial fibration by 4-planes, the product of the standard Lefschetz fibration of $\mathbb C^3$ with a line, and the Harvey-Lawson…
McLean proved that the moduli space of coassociative deformations of a compact coassociative 4-submanifold C in a G_2-manifold (M,phi,g) is a smooth manifold of dimension equal to b^2_+(C). In this paper, we show that the moduli space of coassociative deformations of a noncompact, asymptotically cylindrical coassociative 4-fold C in an asymptotically cylindrical G_2-manifold (M,phi,g) is also a smooth manifold. Its dimension is the dimension of the positive subspace of the image of H^2_cs(C,R) in H^2(C,R).
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.
This paper is a follow-up to an earlier paper math.DG/0410260 on desingularizations of Calabi-Yau 3-folds with a conical singularity. In math.DG/0410260 we study Calabi-Yau 3-folds M_0 with a conical singularity at x modelled on some Calabi-Yau cone V, and construct a desingularization of M_0 by gluing in an Asymptotically Conical (AC) Calabi-Yau 3-fold Y to M_0 at x. In this paper, we shall investigate a similar desingularization problem on special Lagrangian (SL) 3-folds in the corresponding Calabi-Yau 3-folds. More precisely, suppose M_0 is now a Calabi-Yau 3-fold with finitely many conical…
We study Calabi-Yau 3-folds M_0 with a conical singularity x modelled on a Calabi-Yau cone V. We construct desingularizations of M_0, obtaining a 1-parameter family of compact, nonsingular Calabi-Yau 3-folds which has M_0 as the limit. The way we do is to choose an Asymptotically Conical Calabi-Yau 3-fold Y modelled on the same cone V, and then glue into M_0 at x after applying a homothety to Y. We then get a 1-parameter family of nearly Calabi-Yau 3-folds M_t depending on a small real variable t. For sufficiently small t, we show that the nearly Calabi-Yau structures on M_t can be deformed to…
Every closed, oriented, real analytic Riemannian 3-manifold can be isometrically embedded as a special Lagrangian submanifold of a Calabi-Yau 3-fold, even as the real locus of an antiholomorphic, isometric involution. Every closed, oriented, real analytic Riemannian 4-manifold whose bundle of self-dual 2-forms is trivial can be isometrically embedded as a coassociative submanifold in a G_2-manifold, even as the fixed locus of an anti-G_2 involution. These results, when coupled with McLean's analysis of the moduli spaces of such calibrated submanifolds, yield a plentiful supply of examples of c…
We give a complete description of the behaviour of Calabi-Yau instantons and monopoles with an $SU(2)^2$-symmetry, on Calabi-Yau 3-folds with asymptotically conical geometry and $SU(2)^2$ acting with co-homogeneity one. We consider gauge theory on the smoothing and the small resolution of the conifold, and on the canonical bundle of $\mathbb{CP}^1 \times \mathbb{CP}^1$, with their known asymptotically conical co-homogeneity one Calabi-Yau metrics, and find new one-parameter families of invariant instantons. We also entirely classify the relevant moduli-spaces of instantons and monopoles satisf…
In an earlier paper, we proved that given an asymptotically cylindrical G_2-manifold M with a Calabi-Yau boundary X, the moduli space of coassociative deformations of an asymptotically cylindrical coassociative 4-fold C in M with a fixed special Lagrangian boundary L in X is a smooth manifold of dimension dim(V_+), where V_+ is the positive subspace of the image of H^2_{cs}(C,R) in H^2(C,R). In order to prove this we used the powerful tools of Fredholm Theory for noncompact manifolds which was developed by Lockhart and McOwen, and independently by Melrose. In this paper, we extend our result t…
In this article we study the deformation theory of conically singular Cayley submanifolds. In particular, we prove a result on the expected dimension of a moduli space of Cayley deformations of a conically singular Cayley submanifold. Moreover, when the Cayley submanifold is a two-dimensional complex submanifold of a Calabi--Yau four-fold we show by comparing Cayley and complex deformations that in this special case the moduli space is a smooth manifold. We also perform calculations of some of the quantities discussed for some examples.
This is the last in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with this paper. We survey the major results of the previous four papers, giving brief explanations of the proofs. We apply the results to describe the boundary of a moduli space of compact, nonsingular SL m-folds N in M. We prove the…
This is the first in a series of five papers math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272, which surveys the series, gives examples, and applies the results to prove some conjectures. This first paper lays the foundations for the series, giving definitions and proving auxiliary results in symp…
We develop a powerful new analytic method to construct complete non-compact G2-manifolds, i.e. Riemannian 7-manifolds (M,g) whose holonomy group is the compact exceptional Lie group G2. Our construction starts with a complete non-compact asymptotically conical Calabi-Yau 3-fold B and a circle bundle M over B satisfying a necessary topological condition. Our method then produces a 1-parameter family of circle-invariant complete G2-metrics on M that collapses to the original Calabi-Yau metric on the base B as the parameter converges to 0. The G2-metrics we construct have controlled asymptotic ge…
The exceptional holonomy groups are G2 in 7 dimensions, and Spin(7) in 8 dimensions. Riemannian manifolds with these holonomy groups are Ricci-flat. This is a survey paper on exceptional holonomy, in two parts. Part I introduces the exceptional holonomy groups, and explains constructions for compact 7- and 8-manifolds with holonomy G2 and Spin(7). The simplest such constructions work by using techniques from complex geometry and Calabi-Yau analysis to resolve the singularities of a torus orbifold T^7/G or T^8/G, for G a finite group preserving a flat G2 or Spin(7)-structure on T^7 or T^8. Ther…
This is the fourth in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0303272 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and applies the results to prove some conjectures. The first paper math.DG/0211294 studied the regularity of X near its singular points, and the second m…
This is the third in a series of five papers math.DG/0211294, math.DG/0211295, math.DG/0302356, math.DG/0303272 studying compact special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and applies the results to prove some conjectures. The first two papers math.DG/0211294, math.DG/0211295 studied the regularity of X near its singular poi…
This is the second in a series of five papers math.DG/0211294, math.DG/0302355, math.DG/0302356, math.DG/0303272 studying special Lagrangian submanifolds (SL m-folds) X in (almost) Calabi-Yau m-folds M with singularities x_1,...,x_n locally modelled on special Lagrangian cones C_1,...,C_n in C^m with isolated singularities at 0. Readers are advised to begin with the final paper math.DG/0303272 which surveys the series, gives examples, and proves some conjectures. In this paper we study the deformation theory of compact SL m-folds X in M with conical singularities. We define the moduli space M_…
We survey what is known about singularities of special Lagrangian submanifolds (SL m-folds) in (almost) Calabi-Yau manifolds. The bulk of the paper summarizes the author's five papers math.DG/0211294, math.DG/0211295, math.DG/0302355, math.DG/0302356, math.DG/0303272 on SL m-folds X with isolated conical singularities. That is, near each singular point x, X is modelled on an SL cone C in C^m with isolated singularity at 0. We also discuss directions for future research, and give a list of open problems.
We construct infinitely many new 1-parameter families of simply connected complete noncompact G_2-manifolds with controlled geometry at infinity. The generic member of each family has so-called asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G_2-ma…
