Related papers: Constructing exact Lagrangian immersions with few …

We study minimal immersions of closed surfaces (of genus $g \ge 2$) in hyperbolic 3-manifolds, with prescribed data $(\sigma, t\alpha)$, where $\sigma$ is a conformal structure on a topological surface $S$, and $\alpha dz^2$ is a…

Let $M$ be an $n$-dimensional Lagrangian submanifold of a complex space form. We prove a pointwise inequality $$\delta(n_1,\ldots,n_k) \leq a(n,k,n_1,\ldots,n_k) \|H\|^2 + b(n,k,n_1,\ldots,n_k)c,$$ with on the left hand side any…

We construct smooth families of compact special Lagrangian submanifolds embedded in some toric hyper-K\"ahler manifolds, which never become holomorphic Lagrangian submanifolds via any hyper-K\"ahler rotations. These families converge to…

A self-transverse immersion of a smooth manifold M^{k+2} in R^{2k+2} has a double point self-intersection set which is the image of an immersion of a smooth surface, the double point self-intersection surface. We prove that this surface may…

The simple loop conjecture for 3-manifolds states that every 2-sided immersion of a closed surface into a 3-manifold is either injective on fundamental groups or admits a compression. This can be viewed as a generalization of the Loop…

We show that zero-Maslov class Lagrangian self-expanders in C^n which are asymptotic to a pair of planes intersecting transversely are locally unique if n>2 and unique if n=2.

The branched deformations of immersed compact special Lagrangian submanifolds are studied in this paper. If there exists a nondegenerate $\mathbb{Z}_2$ harmonic 1-form over a special Lagrangian submanifold $L$, we construct a family of…

In this note we provide explicit constructions of exact Lagrangian embeddings of tori and Klein bottles inside the symplectisation of an overtwisted contact three-manifold. Note that any closed exact Lagrangian in the symplectisation is…

A theory of transversely oriented spun-normal immersed surfaces in ideally triangulated 3--manifolds is developed in this paper, including linear functionals determining the boundary curves, Euler characteristic and homology class of these…

The paths on the {\bf R$^3$} real Euclidean manifold are defined as 2-dimensional simplicial strips; points are replaced by 2-simplexes and the orbits of the action of a one discrete-parameter group on the base manifold becomes a convex…

This paper shows that there are symplectic four-manifolds M with the following property: a single isotopy class of smooth embedded two-spheres in M contains infinitely many Lagrangian submanifolds, no two of which are isotopic as Lagrangian…

We construct families of imaginary special Lagrangian cylinders near transverse Maslov index $0$ or $n$ intersection points of positive Lagrangian submanifolds in a general Calabi-Yau manifold. Hence, we obtain geodesics of open positive…

We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic four-manifolds: the symplectic vector space $\mathbb{R}^4$, the projective plane $\mathbb{C}P^2$, and the monotone…

n this paper we define an invariant of a pair of 6 dimensional symplectic %optional manifold with vanishing 1st Chern class and its Lagrangian submanifold with vanishing Maslov index. This invariant is a function on the set of the path…

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…

Sz\H ucs proved in 2000 that the $r$-tuple-point manifold of a generic immersion is cobordant to the $\Sigma^{1_{r-1}}$-point manifold of its generic projection. Here we slightly extend this by showing that the natural mappings of these…

We study the topology of Lagrangian submanifolds in standard symplectic vector spaces $\mathbb{C}^n$ using ideas from open-closed string topology. Specifically, for a closed, oriented, spin Lagrangian $L$, we construct a (possibly curved)…

Almost para-Hermitian manifold it is manifold equipped with almost para-complex structure and compatible pseudo-metric of neutral signature. It is considered a class of immersions of almost para-Hermitian manifolds into almost…

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…

An immersion of a smooth $n$-dimensional manifold $M \to \mathbb{R}^q$ is called totally nonparallel if, for every distinct $x, y \in M$, the tangent spaces at $f(x)$ and $f(y)$ contain no parallel lines. Given a manifold $M$, we seek the…