相关论文: Isotopies of high genus Lagrangian surfaces

It is known that any closed, exact Lagrangian in the cotangent bundle of a closed, smooth manifold is of the same homotopy type as the zero section. In this paper, we give a Fukaya-theoretic proof of this fact for the sphere and torus to…

In this paper, we prove that every homogeneous Landsberg surface has isotropic flag curvature. Using this special form of the flag curvature, we prove a rigidity result on homogeneous Landsberg surface. Indeed, we prove that every…

Given a closed exact Lagrangian in the cotangent bundle of a closed smooth manifold, we prove that the projection to the base is a simple homotopy equivalence.

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 prove that for any element in the $\gamma$-completion of the space of smooth compact exact Lagrangian submanifolds of a cotangent bundle, if its $\gamma$-support is a smooth Lagrangian submanifold, then the element itself is a smooth…

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 prove that if two conformal embeddings between Riemann surfaces with finite topology are homotopic, then they are isotopic through conformal embeddings. Furthermore, we show that the space of all conformal embeddings in a given homotopy…

We show that every topological surface lamination of a 3-manifold M is isotopic to one with smoothly immersed leaves. This carries out a project proposed by Gabai in [Problems in foliations and laminations, AMS/IP Stud. Adv. Math. 2.2…

Any two homologous surfaces of the same genus embedded in a smooth 4-manifold X with simply-connected complements are shown to be smoothly isotopic in the connected sum of X and the product of a 2-sphere with itself, if the surfaces are…

We present a new and simpler proof of the fact that any Lagrangian $\mathbb{R}P^2$ in $T^*\mathbb{R}P^2$ is Hamiltonian isotopic to the zero section. Our proof mirrors the one given by Li and Wu for the Hamiltonian uniqueness of Lagrangians…

The manifold M being compact and connected, we prove that every submanifold of its cotangent bundle that is Hamiltonianly isotopic to the zero-section and that is invariant by a Tonelli flow is a graph.

We find an infinite set of eigenfunctions for the Laplacian with respect to a flat metric with conical singularities and acting on degree zero bundles over special Riemann surfaces of genus greater than one. These special surfaces…

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…

An exact Lagrangian submanifold $L \subset X^{2n}$ in a Weinstein sector is called a nearby Lagrangian cocore if it avoids all Lagrangian cocores and is equal to a shifted Lagrangian cocore at infinity. Let $k$ be the dimension of the core…

We prove that all Lagrangian spheres in S^2 x S^2 are Hamiltonian isotopic. The proof uses various properties of holomorphic curves in symplectic manifolds with cylindrical ends which were recently developed in connection with the…

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a…

We study 2-dimensional submanifolds of the space ${\mathbb{L}}({\mathbb{H}}^3)$ of oriented geodesics of hyperbolic 3-space, endowed with the canonical neutral K\"ahler structure. Such a surface is Lagrangian iff there exists a surface in…

We show that for $n\geq 2$ there exists an exact Lagrangian submanifold $L$ in the cotangent bundle $T^*\mathbb{T}^n$ of the $n$-dimensional torus $\mathbb{T}^n$ such that $L$ is symplectically but not Hamiltonian isotopic to the zero…

The purpose of this paper is to present some results on the existence of homologous, nonisotopic symplectic or lagrangian surfaces embedded in a simply connected symplectic 4-dimensional manifold.

Let $G$ be a compact connected Lie group with $\pi_1(G)\cong\mathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of…