Related papers: On the Lifts of Minimal Lagrangian Submanifolds

We address the study of some curvature equations for distinguished submanifolds in para-K\"ahler geometry. We first observe that a para-complex submanifold of a para-K\"ahler manifold is minimal. Next we describe the extrinsic geometry of…

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…

We study minimal surfaces in generic sub-Riemannian manifolds with sub-Riemannian structures of co-rank one. These surfaces can be defined as the critical points of the so-called {\it horizontal} area functional associated to the canonical…

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…

An isometric immersion $f:M^n\to \tilde M^n$ from a Riemannian $n$-manifold $M^n$ into a K\"ahler $n$-manifold $\tilde M^n$ is called {\it Lagrangian} if the complex structure $J$ of the ambient manifold $\tilde M^n$ interchanges each…

We consider a general 4n-dimensional quaternionic Kahler geometry with a free action of the torus T^(n+1). The toric action lifts onto the Swann bundle of the quaternionic Kahler space to a tri-holomorphic action that commutes with the…

We give an algebraic characterisation for the triviality of the canonical bundle of a complex supermanifold in terms of a certain Batalin-Vilkovisky superalgebra structure. As an application, we study the Calabi-Yau case, in which an…

We propose two kinds of gauged linear sigma models whose moduli spaces are real eight-dimensional hyperKahler and Calabi-Yau manifolds, respectively. Here, hyperKahler manifolds have sp(2) holonomy in general and are dual to Type IIB…

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…

We prove that a deformation of a hypersurface in a $(n+1)$-dimensional real space form ${\mathbb S}^{n+1}_{p,1}$ induce a Hamiltonian variation of the normal congruence in the space ${\mathbb L}({\mathbb S}^{n+1}_{p,1})$ of oriented…

We describe several families of Lagrangian submanifolds in the complex Euclidean space which are H-minimal, i.e. critical points of the volume functional restricted to Hamiltonian variations. We make use of various constructions involving…

Let $\pi : E\to M$ be a smooth fiber bundle whose total space is a symplectic manifold and whose fibers are Lagrangian. Let $L$ be an embedded Lagrangian submanifold of $E$. In the paper we address the following question: how can one…

Let $X$ be a K3 surface and let $\text{Spl}(r;c_1,c_2)$ be the moduli space of simple sheaves on $X$ of fixed rank $r$ and Chern classes $c_1$ and $c_2$. Under suitable assumptions, to a pair $(F,W)$ (respectively, $(F,V)$) where $F\in…

Let $(M,g)$ be a Riemannian manifold, $L(M)$ be its frame bundle, $O(M)$ its orthonormal frame bundle. For a distribution $D$ on $M$ we define a subbundle $L(D)\subset L(M)$ or $O(D)\subset O(M)$ in a natural way. This allows us to consider…

We construct consistent Kaluza--Klein reductions of D=11 supergravity to four dimensions using an arbitrary seven-dimensional Sasaki--Einstein manifold. At the level of bosonic fields, we extend the known reduction, which leads to minimal…

In their celebrated paper of 1976, Rothschild and Stein prove a lifting procedure that locally reduces to a free nilpotent Lie algebra any family of smooth vector fields $X_1,\dots,X_q$, over a manifold $M$. Then, a large class 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 prove that certain Riemannian manifolds can be isometrically embedded inside Calabi-Yau manifolds. For example we prove that given any real-analytic one parameter family of Riemannian metrics $g_t$ on a 3-dimensional manifold $Y$ with…

Lagrangian submanifolds in strict nearly K\"ahler 6-manifolds are related to special Lagrangian submanifolds in Calabi-Yau 6-manifolds and coassociative cones in $G_2$-manifolds. We prove that the mean curvature of a Lagrangian submanifold…

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…