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.
We find calibrated submanifolds in neck manifolds. Particularly, we obtain a calibrated submanifold in the Lagrangian self-expander constructed by Joyce, Lee and Tsui.
We review some results concerning the deformations of calibrated minimal submanifolds which occur in Riemannian manifolds with special holonomy. The calibrated submanifolds are assumed compact with a non-empty boundary which is constrained to move in a particular fixed submanifold. The results extend McLean's deformation theory previously developed for closed compact submanifolds.
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}.
`Gluing' is a technique of constructing solutions to non-linear (elliptic) partial differential equations such as Yang--Mills equations, minimal surface equations and Einstein equations. Calibrated submanifolds are a certain class of minimal surfaces, and there are various examples of them constructed by the gluing technique. We have existence theorems in that sense, but there seems to have been no uniqueness theory for higher-dimensional ones such as special Lagrangian submanifolds, which we discuss in the present paper.
We provide an introduction to the theory of calibrated submanifolds through the key examples related with special holonomy. We focus on calibrated geometry in Calabi-Yau, G$_2$ and Spin(7) manifolds, and describe fundamental results and techniques in the field.
For an element $Ψ$ in the graded vector space $Ω^*(M, TM)$ of tangent bundle valued forms on a smooth manifold $M$, a $Ψ$-submanifold is defined as a submanifold $N$ of $M$ such that $Ψ_{|N} \in Ω^*(N, TN)$. The class of $Ψ$-submanifolds encompasses calibrated submanifolds, complex submanifolds and all Lie subgroups in compact Lie groups. The graded vector space $Ω^*(M, TM)$ carries a natural graded Lie algebra structure, given by the Frölicher-Nijenhuis bracket $[-,- ]^{FN}$. When $Ψ$ is an odd degree element with $[ Ψ, Ψ]^{FN} =0$, we associate to a $Ψ$-submanifold $N$ a strongly homotopy Li…
By making use of the symplectic reduction and the cohomogeneity method, we give a general method for constructing Hamiltonian minimal submanifolds in Kaehler manifolds with symmetries. As applications, we construct infinitely many nontrivial complete Hamiltonian minimal submanifolds in CP^n and C^n.
We show a method to construct a special Lagrangian submanifold L' from a given special Lagrangian submanifold L in a Calabi-Yau manifold with the use of generalized perpendicular symmetries. We use moment maps of the actions of Lie groups, which are not necessarily abelian. By our method, we construct some non-trivial examples in the cotangent bundles of the spheres which are non-flat Calabi-Yau manifolds equipped with the Stenzel metrics.
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…
The generalized Feix--Kaledin construction shows that c-projective $2n$-manifolds with curvature of type $(1,1)$ are precisely the submanifolds of quaternionic $4n$-manifolds which are fixed points set of a special type of quaternionic $S^1$ action $v$. In this paper, we consider this construction in the presence of infinitesimal symmetries of the two geometries. First, we prove that the submaximally symmetric c-projective model with type $(1,1)$ curvature is a submanifold of a submaximally symmetric quaternionic model, and show how this fits into the construction. We give conditions for when …
Let $M$ be a Fano manifold equipped with a Kähler form $ω\in 2πc_1(M)$ and $K$ a connected compact Lie group acting on $M$ as holomorphic isometries. In this paper, we show the minimality of a $K$-invariant Lagrangian submanifold $L$ in $M$ w.r.t. a globally conformal Kähler metric is equivalent to the minimality of the reduced Lagrangian submanifold $L_0=L/K$ in a Kähler quotient $M_0$ w.r.t. the Hsiang-Lawson metric. Furthermore, we give some examples of Kähler reductions by using a circle action obtained from a cohomogenenity one action on a Kähler-Einstein manifold of positive Ricci curvat…
In this paper we will investigate torus actions on complete manifolds with calibrations. For Calabi-Yau manifolds M^2n with a Hamiltonian structure-preserving k-torus action we show that any symplectic reduction has a natural holomorphic volume form. Moreover Special Lagrangian (SLag) submanifolds of the reduction lift to SLag submanifolds of M, invariant under the torus action. If k=n-1 and the first cohomology of M is trivial, then we prove that M is a fibration with generic fiber being a SLag submanifold. As an application we will see that crepant resolutions of singularities of a finite Ab…
Anisotropic calibrations, Fueter maps and mirror symmetry
Beijing Institute of Mathematical Sciences and Applications（北京国际数学研究中心）；Department of Mathematics, Osaka Metropolitan University（大阪公立大学数学系）；Department of Mathematics, University of Torino（都灵大学数学系）
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…
Austere submanifolds in Euclidean space were introduced by Harvey and Lawson in connection with their study of calibrated geometries. The algebraic possibilities for second fundamental forms of 4-dimensional austere submanifolds were classified by Bryant, into three types which we label A, B, and C. In this paper, we show that type A submanifolds correspond exactly to real Kahler submanifolds, we construct new examples of such submanifolds in R^6 and R^10, and we obtain classification results on submanifolds of types B and C with maximal second fundamental forms.
Let H be the hyperbolic space of dimension n+1. A geodesic foliation of H is given by a smooth unit vector field on H all of whose integral curves are geodesics. Each geodesic foliation of H determines an n-dimensional submanifold M of the 2n-dimensional manifold L of all the oriented geodesics of H (up to orientation preserving reparametrizations). The space L has a canonical split semi-Riemannian metric induced by the Killing form of the isometry group of H. Using a split special Lagrangian calibration, we study the volume maximization problem for a certain class of geometrically distinguish…
In this paper we investigate the geometry of Calibrated submanifolds and study relations between their moduli-space and geometry of the ambient manifold. In particular for a Calabi-Yau manifold we define Special Lagrangian submanifolds for any Kahler metric on it. We show that for a choice of Kahler metric the Borcea-Voisin threefold has a fibration structure with generic fiber being a Special Lagrangian torus. Moreover we construct a mirror to this fibration. Also for any closed G_2 form on a 7-manifold we study coassociative submanifolds and we show that one example of a G_2 manifold constru…
For Hitchin's generalised geometries we introduce and analyse the concept of a structured submanifold which encapsulates the classical notion of a calibrated submanifold. Under a suitable integrability condition on the ambient geometry, these generalised calibrated cycles minimise a functional occurring as D-brane energy in type II string theories, involving both so-called NS-NS- and R-R-fields. Further, we investigate the behaviour of calibrated cycles under T-duality and construct non-trivial examples.
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 $S^{7}$.
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…
We revisit McLean's second variation formulas for calibrated submanifolds in exceptional geometries, and correct his formulas concerning associative submanifolds and Cayley submanifolds, using a unified treatment based on the (relative) calibration method and Harvey-Lawson's identities.
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…
This is the first in a series of papers on special Lagrangian submanifolds in C^m. We study special Lagrangian submanifolds in C^m with large symmetry groups, and give a number of explicit constructions. Our main results concern special Lagrangian cones in C^m invariant under a subgroup G in SU(m) isomorphic to U(1)^{m-2}. By writing the special Lagrangian equation as an o.d.e. in G-orbits and solving the o.d.e., we find a large family of distinct, G-invariant special Lagrangian cones on T^{m-1} in C^m. These examples are interesting as local models for singularities of special Lagrangian subm…
On a Riemannian manifold $\bar{M}^{m+n}$ with an $(m+1)$-calibration $Ω$, we prove that an $m$-submanifold $M$ with constant mean curvature $H$ and calibrated extended tangent space $\mathbb{R}H\oplus TM$ is a critical point of the area functional for variations that preserve the enclosed $Ω$-volume. This recovers the case described by Barbosa, do Carmo and Eschenburg, when $n=1$ and $Ω$ is the volume element of $\bar{M}$. To the second variation we associate an $Ω$-Jacobi operator and define $Ω$-stablility. Under natural conditions, we prove that the Euclidean $m$-spheres are the unique $Ω$-s…
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…
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…
Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivial bundle of self-dual 2-forms, may be isometrically embedded into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show that Bochner's Theorem on real analytic approximation of smooth differential forms, can be obtained using real algebraic tools developed by A…
In this paper we introduce and study the notion of plurisubharmonic functions in calibrated geometry. These functions generalize the classical plurisubharmonic functions from complex geometry and enjoy many of their important properties. Moreover, they exist in abundance whereas the corresponding pluriharmonics are generally quite scarce. A number of the results established in complex analysis via plurisubharmonic functions are extended to calibrated manifolds. This paper investigates, in depth, questions of: pseudo-convexity and cores, positive phi-currents, Duval-Sibony Duality, and boundari…
In this paper, we show that the calibrated method can also be used to detect indefinite minimal Lagrangian submanifolds in $C_k^m$. We introduce the notion of indefinite special Lagrangian submanifolds in $C_k^m$ and generalize the well-known work of Harvey-Lawson to the indefinite case.
Given $(\bar{M},Ω)$ a calibrated Riemannian manifold with a parallel calibration of rank $m$, and $M^m$ an immersed orientable submanifold with parallel mean curvature $H$ we prove that if $\cos θ$ is bounded away from zero, where $θ$ is the $Ω$-angle of $M$, and if $M$ has zero Cheeger constant, then $M$ is minimal. In the particular case $M$ is complete with $Ricc^M\geq 0$ we may replace the boundedness condition on $\cos θ$ by $\cos θ\geq Cr^{-β}$, when $r\to +\infty$, where $ 0\leqβ<1 $ and $C > 0$ are constants and $r$ is the distance function to a point in $M$. Our proof is surprisingly …
The geometry of submanifolds is intimately related to the theory of functions and vector bundles. It has been of fundamental importance to find out how those two objects interact in many geometric and physical problems. A typical example of this relation is that the Picard group of line bundles on an algebraic manifold is isomorphic to the group of divisors, which is generated by holomorphic hypersurfaces modulo linear equivalence. A similar correspondence can be made between the K-group of sheaves and the Chow ring of holomorphic cycles. There are two more very recent examples of such a relat…
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,…
We extend the "bundle constructions" of calibrated submanifolds, due to Harvey--Lawson in the special Lagrangian case, and to Ionel--Karigiannis--Min-Oo in the cases of exceptional calibrations, by "twisting" the bundles by a special (harmonic, holomorphic, parallel) section of a complementary bundle. The existence of such deformations shows that the moduli space of calibrated deformations of these "calibrated subbundles" includes deformations which destroy the linear structure of the fibre.
In this paper we give a survey of various results about the topology of oriented Grassmannian bundles related to the exceptional Lie group G_2. Some of these results are new. We give self-contained proofs here. One often encounters these spaces when studying submanifolds of manifolds with calibrated geometries. For the sake of completeness we decided to collect them here in a self-contained way to be easily accessible for future usage in calibrated geometry. As an application we deduce existence of certain special 3 and 4 dimensional submanifolds of G_2 manifolds with special properties, which…
We establish a link between the study of completely integrable systems of partial differential equations and the study of generic submanifolds in C^n. Using the recent developments of Cauchy-Riemann geometry we provide the set of symmetries of such a system with a Lie group structure. Finally we determine the precise upper bound of the dimension of this Lie group for some specific systems of partial differential equations.
E. Cartan proved that conformally flat hypersurfaces in S^{n+1} for n>3 have at most two distinct principal curvatures and locally envelop a one-parameter family of (n-1)-spheres. We prove that the Gauss-Codazzi equation for conformally flat hypersurfaces in S^4 is a soliton equation, and use a dressing action from soliton theory to construct geometric Ribaucour transforms of these hypersurfaces. We describe the moduli of these hypersurfaces in S^4 and their loop group symmetries. We also generalise these results to conformally flat n-immersions in (2n-2)-spheres with flat normal bundle and co…
We introduce the equation of n-dimensional totally geodesic submanifolds of a manifold E as a submanifold of the second order jet space of n-dimensional submanifolds of E. Next we study the geometry of n-Grassmannian equivalent connections, that is linear connections without torsion admitting the same equation of n-dimensional totally geodesic submanifolds. We define the n-Grassmannian structure as the equivalence class of such connections, recovering for n=1 the case of theory of projectively equivalent connections. By introducing the equation of parametrized n-dimensional totally geodesic su…
In this paper we introduce the area of 2-ruled 4-folds in R^n (n=7 or 8), that is, submanifolds M of R^n that admit a fibration over some 2-fold Sigma such that each fibre is an affine 2-plane in R^n. This is motivated by the paper math.DG/0012060 by Joyce on ruled special Lagrangian 3-folds in C^3 and the work of the author in math.DG/0401123 on ruled associative 3-folds in R^7. We say that a 2-ruled 4-fold M is r-framed if we are given an oriented basis for each fibre in a smooth manner, and in such circumstances we may write M in terms of orthogonal smooth maps phi_1,phi_2:Sigma-->S^(n-1) a…
This paper gives a leisurely introduction to Calabi-Yau manifolds and special Lagrangian submanifolds from the differential geometric point of view, followed by a survey of recent results on singularities of special Lagrangian submanifolds, and their application to the SYZ Conjecture. It is aimed at graduate students in Geometry, String Theorists, and others wishing to learn the subject, and is designed to be fairly self-contained. It is based on lecture courses given at Nordfjordeid, Norway and MSRI, Berkeley in June and July 2001. We introduce Calabi-Yau m-folds via holonomy groups, Kahler g…
We study the equation E_fc of flat connections in a fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation E are in 1-1 correspondence with morphisms f: E\to E_fc, where E and E_fc are treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDE. In particular, the Lax pairs of the self-dual Yang-Mills equations and their reductions are of this type. With each flat repre…
This thesis is devoted to algorithmic aspects of the implementation of Cartan's moving frame method to the problem of the equivalence of submanifolds under a Lie group action. We adopt a general definition of a moving frame as an equivariant map from the space of submanifolds to the group itself and introduce two algorithms, which simplify the construction of such maps. The first algorithm is applicable when the group factors as a product of two subgroups $G=BA$, allowing us to use moving frames and differential invariants for the groups $A$ and $B$ in order to construct a moving frame and dif…
