Related papers: Minimal submanifolds of Kaehler-Einstein manifolds…

A closed connected oriented Riemannian manifold $N$ with non-vanishing Euler characteristic, non-negative curvature operator and $0< 2\text{Ric}_N<\text{scal}_N$ is area-rigid in the sense that any area non-increasing spin map $f\colon M\to…

We prove a universal embedding theorem for flag manifolds: every flag manifold admits a holomorphic isometric embedding into an irreducible classical flag manifold. This result generalizes the classical celebrated embedding theorems of…

A submanifold of a Riemannian manifold is called a parallel submanifold if its second fundamental form is parallel with respect to the van der Waerden-Bortolotti connection. From submanifold point of view, parallel submanifolds are the…

We consider exact Lagrangian submanifolds in cotangent bundles. Under certain additional restrictions (triviality of the fundamental group of the cotangent bundle, and of the Maslov class and second Stiefel-Whitney class of the Lagrangian…

We give necessary and sufficient conditions for a Lagrangian submanifold of a K\"ahler manifold to be biharmonic. Furthermore, we classify biharmonic PNMC Lagrangian submanifolds in the complex space forms.

We uncover the lowest order differential invariants of Lagrangian submanifolds under affine symplectic maps, and find out what happens when they are constant.

Reflection in a line in Euclidean 3-space defines an almost paracomplex structure on the space of all oriented lines, isometric with respect to the canonical neutral Kaehler metric. Beyond Euclidean 3-space, the space of oriented geodesics…

An almost K\"ahler structure on a symplectic manifold $(N, \omega)$ consists of a Riemannian metric $g$ and an almost complex structure $J$ such that the symplectic form $\omega$ satisfies $\omega(\cdot, \cdot)=g(J(\cdot), \cdot)$. Any…

Let the circle act holomorphically on a compact K\"ahler manifold $M$ of complex dimension $n$ with moment map $\phi\colon M\to\R$. Assume the critical set of $\phi$ consists of 3 connected components, the extrema being isolated points. We…

We compare Hofer's geometries on two spaces associated with a closed symplectic manifold M. The first space is the group of Hamiltonian diffeomorphisms. The second space L consists of all Lagrangian submanifolds of $M \times M$ which are…

The Riemannian product of two hyperbolic planes of constant Gaussian curvature -1 has a natural K\"ahler structure. In fact, it can be identified with the complex hyperbolic quadric of complex dimension two. In this paper we study…

A contact structure on a complex manifold M is a corank 1 subbundle F of T(M) such that the bilinear form on F with values in the quotient line bundle L=T(M)/F deduced from the Lie bracket of vector fields is everywhere non-degenerate. In…

Let L be a Lagrangian submanifold of a pseudo- or para-K\"ahler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect…

We prove that for any given compact Riemannian manifold $N$ of dimension $n+1 \geq 3$ and any non-negative Lipschitz function $g$ on $N$, there exists a quasi-embedded, boundaryless hypersurface $M \subset N,$ of class $C^{2, \alpha}$ for…

For an arbitrary submanifold $M \subset \mathbb{C}P^n$ we determine conditions under which it is austere, i.e., the normal bundle of $M$ is special Lagrangian with respect to Stenzel's Ricci-flat K\"ahler metric on $T\mathbb{C}P^n$. We also…

We show that the Hamiltonian Lagrangian monodromy group, in its homological version, is trivial for any weakly exact Lagrangian submanifold of a symplectic manifold. The proof relies on a sheaf approach to Floer homology given by a relative…

This article explores special Lagrangian submanifolds in $\mathbb{CP}^3$, viewed as a nearly K\"ahler manifold, from two different perspectives. Intrinsically, using a moving frame set-up, and extrinsically, using $\mathrm{SU}(2)$…

Having fixed a Kaehler class and the unique corresponding hyperkaehler metric, we prove that all special Lagrangian submanifolds of an irreducible symplectic 4-fold X are bi-Lagrangian and that they are obtained by complex submanifolds via…

We study non-totally geodesic Lagrangian submanifolds of the nearly K\"ahler $\mathbb{S}^3 \times \mathbb{S}^3$ for which the projection on the first component is nowhere of maximal rank. We show that this property can be expressed in terms…

We study exact Lagrangian immersions with one double point of a closed orientable manifold K into n-complex-dimensional Euclidean space. Our main result is that if the Maslov grading of the double point does not equal 1 then K is homotopy…