Complex anti-self-dual instantons and Cayley submanifolds

Deformations of Asymptotically Cylindrical Cayley Submanifolds

We study the deformations of an asymptotically cylindrical Cayley submanifold inside an asymptotically cylindrical Spin(7)-manifold. We prove an index formula for the operator of Dirac type that arises as the linearisation of the deformation map and show that if the Spin(7)-structure is generic, then there are no obstructions, and hence the moduli space is a smooth finite-dimensional manifold whose dimension is equal to the index of the operator of Dirac type. We further construct examples of asymptotically cylindrical Cayley submanifolds inside the asymptotically cylindrical Riemannian 8-mani…

Totally geodesic submanifolds of Damek-Ricci spaces and Einstein hypersurfaces of the Cayley projective plane

We classify totally geodesic submanifolds of Damek-Ricci spaces and show that they are either homogeneous (such submanifolds are known to be "smaller" Damek-Ricci spaces) or isometric to rank-one symmetric spaces of negative curvature. As a by-product, we obtain that a totally geodesic submanifold of any known harmonic manifold is by itself harmonic. We prove that the Cayley hyperbolic plane admits no Einstein hypersurfaces and that the only Einstein hypersurfaces in the Cayley projective plane are geodesic spheres of a particular radius; this completes the classification of Einstein hypersurf…

$G_2$-instantons, associative submanifolds, and Fueter sections

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.

Deformations of homogeneous associative submanifolds in nearly parallel $G_{2}$-manifolds

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 $S^{7}$.

锥形奇异 Cayley 子流形 III：纤维化

Conically singular Cayley submanifolds III: Fibrations

本文使用粘合方法构造非平凡 Cayley 纤维化，证明了弱纤维化和通常意义下 Cayley 纤维化的稳定性，并在扭曲连通和 $G_2$ 流形与圆的乘积上构造了 Cayley 及共结合纤维化的实例，完成了 Kovalev 的长期计划。

Cayley deformations of compact complex surfaces

In this article, we consider Cayley deformations of a compact complex surface in a Calabi--Yau four-fold. We will study complex deformations of compact complex submanifolds of Calabi--Yau manifolds with a view to explaining why complex and Cayley deformations of a compact complex surface are the same. We in fact prove that the moduli space of complex deformations of any compact complex embedded submanifold of a Calabi--Yau manifold is a smooth manifold.

Cayley submanifolds of Calabi-Yau 4-folds

Our main results are: (1) The complex a Lagrangian points of a non-complex Lagrangian $2n$-dimensional submanifold $F:M\ra N$, immersed with parallel mean curvature and with equal Kaehler angles into a Kaehler-Einstein manifold $(N,J,g)$ of complex dimension $2n$, are zeros of finite order of $\sin^2θ$ and $\cos^2θ$ respectively, where $θ$ is the common $J$-Kaelher angle. (2) If $M$ is a Cayley submanifold of a Calabi-Yau (CY) manifold $N$ of complex dimension 4, then $\bigwedge^2_+NM$ is naturally isomorphic to $\bigwedge^2_+TM$. (3) If $N$ is Ricci-flat (not necessarily CY) and $M$ is a Cayl…

Calibrated Submanifolds of R^7 and R^8 with Symmetries

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.

接触阶与直纹子流形

Order of contact and ruled submanifolds

本研究借助初等几何测度论，推广了关于密切性与直纹子流形的Monge-Cayley-Salmon定理。

The Mean Curvature of First-Order Submanifolds in Exceptional Geometries with Torsion

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.

紧型秩一对称空间中子流形的第二特征值估计与谱刚性

Second Eigenvalue Estimates and Spectral Rigidity for Submanifolds of Compact Rank-One Symmetric Spaces

该研究针对紧射影空间及凯莱射影平面上的闭子流形，建立薛定谔算子第二特征值的上界，推导普适估计，证明等式对应全脐子流形，确定达界的全测地例子。

Instantons in G2 manifolds from J-holomorphic curves in coassociative submanifolds

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.

Instantons and branes in manifolds with vector cross product

In this paper we study the geometry of manifolds with vector cross product and its complexification. First we develop the theory of instantons and branes and study their deformations. For example they are (i) holomorphic curves and Lagrangian submanifolds in symplectic manifolds and (ii) associative submanifolds and coassociative submanifolds in G_2-manifolds. Second we classify Kahler manifolds with the complex analog of vector cross product, namely they are Calabi-Yau manifolds and hyperkahler manifolds. Furthermore we study instantons, Neumann branes and Dirichlet branes on these manifolds.…

Curvature-adapted submanifolds of symmetric spaces

We study curvature-adapted submanifolds of general symmetric spaces. We generalize Cartan's theorem for isoparametric hypersurfaces of spheres and Wang's classification of isoparametric Hopf hypersurfaces in complex projective spaces to any compact symmetric space. Our second objective is to investigate such hypersurfaces in some specific symmetric spaces. Various classification results in the Cayley projective and hyperbolic planes and in complex two-plane Grassmannians are obtained under some additional assumptions.

${\rm Spin}(7)$-Instantons from evolution equations

In this paper we study ${\rm Spin}(7)$-instantons on asymptotically conical ${\rm Spin}(7)$-orbifolds (and manifolds) obtained by filling in certain squashed $3$-Sasakian $7$-manifolds. We construct a $1$-parameter family of explicit ${\rm Spin}(7)$-instantons. Taking the parameter to infinity, the family (a) bubbles off an ASD connection in directions transverse to a certain Cayley submanifold $Z$, (b) away from $Z$ smoothly converges to a limit ${\rm Spin}(7)$-instanton that extends across $Z$ onto a topologically distinct bundle, (c) satisfies an energy conservation law for the instantons a…

校准子流形的一个变分刻画

A variational characterization of calibrated submanifolds

该研究通过特定方式改变流形上的校准构造外围度量的特殊变分，证明紧定向嵌入子流形被校准当且仅当外围度量是体积泛函的临界点，推广了殆凯勒情形的已有结果，并应用于Smith映射的变分刻画。

Intersection theory of coassociative submanifolds in G_(2)-manifolds and Seiberg-Witten invariants

We study the problem of counting instantons with coassociative boundary condition in (almost) G_(2)-manifolds. This is analog to the open Gromov-Witten theory for counting holomorphic curves with Lagrangian boundary condition in Calabi-Yau manifolds. We explain its relationship with the Seiberg-Witten invariants for coassociative submanifolds.

Thin instantons in G_2-manifolds and Seiberg-Witten invariants

For two nearby disjoint coassociative submanifolds C and C' in a G_2-manifold, we construct thin instantons with boundaries lying on C and C' from regular J-holomorphic curves in C. We explain their relationship with the Seiberg-Witten invariants for C.

Spin(7) instantons and the Hodge Conjecture for certain abelian four-folds: a modest proposal

The Hodge Conjecture is equivalent to a statement about conditions under which a complex vector bundle on a smooth complex projective variety admits a holomorphic structure. I advertise a class of abelian four-folds due to Mumford where this approach could be tested. I construct explicit smooth vector bundles - which can in fact be constructed in terms of of smooth line bundles - whose Chern characters are given Hodge classes. An instanton connection on these vector bundles would endow them with a holomorphic structure and thus prove that these classes are algebraic. I use complex multiplicati…

Riemannian geometry over different normed division algebra

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…