We study associative submanifolds of the Berger space SO(5)/SO(3) endowed with its homogeneous nearly-parallel G2-structure. We focus on two geometrically interesting classes: the ruled associatives, and the associatives with special Gauss map.
We show that the associative submanifolds ruled by a certain special type of geodesic are in correspondence with pseudo-holomorphic curves in $Gr_2^+(TS^4).$ Using this correspondence, together with a theorem of Bryant on superminimal surfaces in $S^4,$ we prove the existence of infinitely many topological types of compact immersed associative 3-folds in SO(5)/SO(3).
An associative submanifold of the Berger space is said to have special Gauss map if its tangent spaces have non-trivial SO(3)-stabiliser. We classify the associative submanifolds with special Gauss map in the cases where the stabiliser contains an element of order greater than 2. In particular, we find several homogeneous examples of this type.
The stability and the index of compact minimal submanifolds of the Berger spheres $\mathbb{S}^{2n+1}_τ,\, 0<τ\leq 1$, are studied. Unlike the case of the standard sphere ($τ=1$), where there are no stable compact minimal submanifolds, the Berger spheres have stable ones if and only if $τ^2\leq 1/2$. Moreover, there are no stable compact minimal $d$-dimensional submanifolds of $\mathbb{S}^{2n+1}_τ$ when $1 / (d+1) < τ^2 \leq 1$ and the stable ones are classified for $τ^2=1 / (d+1)$ when the submanifold is embedded. Finally, the compact orientable minimal surfaces of $\mathbb{S}^3_τ$ with index …
We prove a Berger type theorem for the normal holonomy group (i.e., the holonomy group of the normal connection) of a full complete complex submanifold of the complex projective space. Namely, if the normal holonomy does not act transitively, then the submanifold is the complex orbit, in the complex projective space, of the isotropy representation of an irreducible Hermitian symmetric space of rank greater or equal to 3. Moreover, we show that for complete irreducible complex submanifolds of the complex Euclidean space the normal holonomy is generic, i.e., it acts transitively on the unit sphe…
We discuss a model for associative submanifolds in $G_{2}$-manifolds with K3 fibrations, in the adiabatic limit. The model involves graphs in a 3-manifold whose edges are locally gradient flow lines. We show that this model produces analogues of known singularity formation phenomena for associative submanifolds.
Associative submanifolds $A$ in nearly parallel $G_2$-manifolds $Y$ are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over $Y$ has the holonomy group contained in ${\rm Spin(7)}$ and the Riemannian cone over $A$ is a Cayley submanifold. Infinitesimal deformations of associative submanifolds were considered by the author. This paper is a continuation of the work. We give a necessary and sufficient condition for an infinitesimal associative deformation to be integrable (unobstructed) to second order explicitly. As an application, we show that the infi…
We give a sufficient condition for an associative submanifold in a G2-manifold to appear as the bubbling locus of a sequence of G2-instantons, related to the existence of a Fueter section of a bundle of ASD instanton moduli spaces over said submanifold.
Let M^7 a manifold with holonomy in G_2, and Y^3 an associative submanifold with boundary in a coassociative submanifold. In [5], the authors proved that M_{X,Y}, the moduli space of its associative deformations with boundary in the fixed X, has finite virtual dimension. Using Bochner's technique, we give a vanishing theorem that forces M_{X,Y} to be locally smooth.
Building on ideas from [DT98; DS11; Wal17; Hay17], we outline a proposal for constructing Floer homology groups associated with a G2-manifold. These groups are generated by associative submanifolds and solutions of the ADHM Seiberg-Witten equations. The construction is motivated by the analysis of various transitions which can change the number of associative submanifolds. We discuss the relation of our proposal to Pandharipande and Thomas' stable pair invariant of Calabi-Yau 3-folds.
A nearly parallel $G_{2}$-manifold $Y$ is a Riemannian 7-manifold whose cone $C(Y) = \mathbb{R}_{>0} \times Y$ has the holonomy group contained in ${\rm Spin(7)}$. In other words, it is a spin 7-manifold with a real Killing spinor. We have a special class of calibrated submanifolds called Cayley submanifolds in $C(Y)$. An associative submanifold in $Y$ is a minimal 3-submanifold whose cone is Cayley. We study its deformations, namely, Cayley cone deformations, explicitly when it is homogeneous in the 7-sphere $S^{7}$.
The squashed 7-sphere $S^{7}$ is a 7-sphere with an Einstein metric given by the canonical variation and its cone $\mathbb{R}^{8} - \{ 0 \}$ has full holonomy ${\rm Spin}(7)$. There is a canonical calibrating 4-form $Φ$ on $\mathbb{R}^{8} - \{ 0 \}$. A minimal 3-submanifold in $S^{7}$ is called associative if its cone is calibrated by $Φ$. In this paper, we classify two types of fundamental associative submanifolds in the squashed $S^{7}$. One is obtained by the intersection with a 4-plane and the other is homogeneous. Then we study their infinitesimal associative deformations and explicitly s…
Let $M$ be a topological $G_2$-manifold. We prove that the space of infinitesimal associative deformations of a compact associative submanifold $Y$ with boundary in a coassociative submanifold $X$ is the solution space of an elliptic problem. For a connected boundary $\partial Y$ of genus $g$, the index is given by $\int_{\partial Y}c_1(ν_X)+1-g$, where $ν_X$ denotes the orthogonal complement of $T\partial Y$ in $TX_{|\partial Y}$ and $c_1(ν_X)$ the first Chern class of $ν_X$ with respect to its natural complex structure. Further, we exhibit explicit examples of non-trivial index.
We obtain inequalities for all Laplace eigenvalues of Riemannian manifolds with an upper sectional curvature bound, whose rudiment version for the first Laplace eigenvalue was discovered by Berger in 1979. We show that our inequalities continue to hold for conformal metrics, and moreover, extend naturally to minimal submanifolds. In addition, we obtain explicit estimates for Laplace eigenvalues of minimal submanifolds in terms of geometric quantities of the ambient space.
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 Kähler 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,…
Let $M^7$ be a smooth manifold equipped with a $G_2$-structure $ϕ$, and $Y^3$ be an closed compact $ϕ$-associative submanifold. In \cite{McL}, R. McLean proved that the moduli space $\bm_{Y,ϕ}$ of the $ϕ$-associative deformations of $Y$ has vanishing virtual dimension. In this paper, we perturb $ϕ$ into a $G_2$-structure $ψ$ in order to ensure the smoothness of $\bm_{Y,ψ}$ near $Y$. If $Y$ is allowed to have a boundary moving in a fixed coassociative submanifold $X$, it was proved in \cite{GaWi} that the moduli space $\bm_{Y,X}$ of the associative deformations of $Y$ with boundary in $X$ has f…
The principal theory of this paper comprises a technique for constructing associative, coassociative and Cayley submanifolds of Euclidean space with symmetries, using first-order ordinary differential equations. Explicit examples of U(1)-invariant associative cones in R^7 and SU(2)-invariant Cayley 4-folds in R^8 are then produced using this method. Further examples of associative 3-folds are presented, which are ruled, and other systems of differential equations defining calibrated submanifolds in R^7 and R^8 are given.
Every compact 3-Sasakian 7-manifold $M$ admits a canonical 2-parameter family of co-closed $\text{G}_2$-structures $φ_{a,b}$ for $a,b > 0$, as well as a foliation by $φ_{a,b}$-associative 3-folds whose leaf space $X$ is a positive quaternion-Kähler 4-orbifold. We prove that associative 3-folds in $(M,φ_{a,b})$ that are ruled by a certain type of geodesic are in correspondence with pseudo-holomorphic curves in the almost-complex 8-manifold $Z \times S^2$, where $Z$ is the twistor space of $X$ equipped with its strict nearly-Kähler structure. As an application, we construct infinitely many topol…
We study deformations of associative submanifolds $Y^3\subset M^7$ of a $G_2$ manifold $M^7$. We show that the deformation space can be perturbed to be smooth, and it can be made compact and zero dimensional by constraining it with an additional equation. This allows us to associate local invariants to associative submanifolds of $M$. The local equations at each associative $Y$ are restrictions of a global equation on a certain associated Grassmann bundle over $ M$.
By analogy with associative and co-associative cases we introduce a class of three-dimensional non-orientable submanifolds, of almost $\mathrm{G}_2-$manifolds, modelled on planes lying in a special $\mathrm{G}_2-$orbit. An application of the Cartan-Kähler theory shows that some three-manifold can be presented in this way. We also classify all the homogeneous ones in $\mathbb{RP}^7$.
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.
By a theorem of Mclean, the deformation space of an associative submanifold Y of an integrable G_2 manifold (M,ϕ) can be identified with the kernel of a Dirac operator D:Ω^{0}(ν) -->Ω^{0}(ν) on the normal bundle νof Y. Here, we generalize this to the non-integrable case, and also show that the deformation space becomes smooth after perturbing it by natural parameters, which corresponds to moving Y through `pseudo-associative' submanifolds. Infinitesimally, this corresponds to twisting the Dirac operator D --> D_{A} with connections A of ν. Furthermore, the normal bundles of the associative sub…
We construct calibrated submanifolds of R^7 and R^8 by viewing them as total spaces of vector bundles and taking appropriate sub-bundles which are naturally defined using certain surfaces in R^4. We construct examples of associative and coassociative submanifolds of R^7 and of Cayley submanifolds of R^8. This construction is a generalization of the Harvey-Lawson bundle construction of special Lagrangian submanifolds of R^{2n}.
We describe a class of compact $G_2$ orbifolds constructed from non-symplectic involutions of K3 surfaces. Within this class, we identify a model for which there are infinitely many associative submanifolds contributing to the effective superpotential of M-theory compactifications. Under a chain of dualities, these can be mapped to F-theory on a Calabi-Yau fourfold, and we find that they are dual to an example studied by Donagi, Grassi and Witten. Finally, we give two different descriptions of our main example and the associative submanifolds as a twisted connected sum.
We provide a significant extension of the twisted connected sum construction of G_2-manifolds, i.e. Riemannian 7-manifolds with holonomy group G_2, first developed by Kovalev; along the way we address some foundational questions at the heart of the twisted connected sum construction. Some of the main contributions of the paper are: (i) We correct, clarify and extend several aspects of the K3 "matching problem" that occurs as a key step in the twisted connected sum construction. (ii) We show that the large class of asymptotically cylindrical Calabi-Yau 3-folds built from semi-Fano 3-folds (a su…
This paper is a continuation of math.DG/0408005. We first construct special Lagrangian submanifolds of the Ricci-flat Stenzel metric (of holonomy SU(n)) on the cotangent bundle of S^n by looking at the conormal bundle of appropriate submanifolds of S^n. We find that the condition for the conormal bundle to be special Lagrangian is the same as that discovered by Harvey-Lawson for submanifolds in R^n in their pioneering paper. We also construct calibrated submanifolds in complete metrics with special holonomy G_2 and Spin(7) discovered by Bryant and Salamon on the total spaces of appropriate bun…
In G2 manifolds, 3-dimensional associative submanifolds (instantons) play a role similar to J-holomorphic curves in symplectic geometry. In [21], instantons in G2 manifolds were constructed from regular J-holomorphic curves in coassociative submanifolds. In this exposition paper, after reviewing the background of G2 geometry, we explain the main ingredients in the proofs in [21]. We also construct new examples of instantons.
We construct a gauge fixed action for topological membranes on $G_2$-manifold such that its bosonic part is the standard membrane theory in a particular gauge. We prove that quantum mechanically the path-integral in this gauge localizes on associative submanifolds. Moreover on $M\times S^1$ the theory naturally reduces to the standard A-model on Calabi-Yau manifold and to a membrane theory localized on special Lagrangian submanifolds. We discuss some properties of topological membrane theory on $G_2$-manifolds. We also generalize our construction to topological $p$--branes on special manifolds…
We develop a unifed theory to study geometry of manifolds with different holonomy groups. They are classified by (1) real, complex, quaternion or octonion number they are defined over and (2) being special or not. Specialty is an orientation with respect to the corresponding normed algebra A. For example, special Riemannian A-manifolds are oriented Riemannian, Calabi-Yau, Hyperkahler and G_2-manifolds respectively. For vector bundles over such manifolds, we introduce (special) A-connections. They include holomorphic, Hermitian Yang-Mills, Anti-Self-Dual and Donaldson-Thomas connections. Simila…
