Related papers: Calibrated Sub-Bundles in Non-Compact Manifolds of…

We study special Lagrangian submanifolds in the Calabi-Yau manifold $T^*S^n$ with the Stenzel metric, as well as calibrated submanifolds in the $\text{G}_2$-manifold $\Lambda^2_-(T^*X)$ $(X^4 = S^4, \mathbb{CP}^2)$ and the…

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…

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…

We show the total space of the canonical line bundle $\mathbb{L}$ of a Kahler-Einstein manifold $X^n$ supports integrable $SU(n+1)$ structures, or Calabi-Yau structures. The canonical real line bundle $L \subset \mathbb{L}$ over a minimal…

We construct calibrated submanifolds in Euclidean space invariant under the action of a Lie group $G$. We first demonstrate the method used in this paper by reproducing the results about special Lagrangians due to Harvey-Lawson. We then…

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…

We construct new explicit metrics on complete non-compact Riemannian 8-manifolds with holonomy Spin(7). One manifold, which we denote by A_8, is topologically R^8 and another, which we denote by B_8, is the bundle of chiral spinors over…

Coassociative submanifolds are 4-dimensional calibrated submanifolds in $G_{2}$-manifolds. In this paper, we construct explicit examples of coassociative submanifolds in $\Lambda^{2}_{-} S^{4}$, which is the complete $G_{2}$-manifold…

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,…

We study collections of exact Lagrangian submanifolds respecting some uniform Riemannian bounds, which we equip with a metric naturally arising in symplectic topology (e.g. the Lagrangian Hofer metric or the spectral metric). We exhibit…

We give a necessary and sufficient condition for a special Lagrangian submanifold in C^n constructed by Lawlor being also special Lagrangian in C^n with the Fubini-Study form.

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.…

We show that two of the Bryant-Salamon G_2-manifolds have a simple topology ; homeomorphic to the complement of some submanifolds of the 7-dimensional sphere. In this connection, we show there exists a complete Ricci-flat (non-flat) metric…

In this Letter we establish a relationship between symmetric SU(2) Yang--Mills instantons and metrics with Spin(7)-holonomy. Our method is based on a slight extension of that of Bryant and Salamon developed to construct explicit manifolds…

This short and fairly informal note is an attempt to explain how methods of homological algebra may be brought to bear on problems in symplectic geometry. We do this by looking at a familiar sample question, which is that of the topology of…

The purpose of this paper is to introduce Harvey-Lawson manifolds and review the construction of certain mirror dual Calabi-Yau submanifolds inside a G_2 manifold. More specifically, given a Harvey-Lawson manifold HL, we explain how to…

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…

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…

For a Riemannian manifold $(N,g)$, we construct a scalar flat metric $G$ in the tangent bundle $TN$. It is locally conformally flat if and only if either, $N$ is a 2-dimensional manifold or, $(N,g)$ is a real space form. It is also shown…

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…