Punctured holomorphic curves and Lagrangian embeddings

A symplectic manifold (M, ω) is a smooth manifold M endowed with a nondegenerate and closed 2-form ω. By Darboux's Theorem such a manifold looks locally like an open set in some R 2n ∼ = C n with the standard symplectic formand so symplectic manifolds have no local invariants. This is in sharp contrast to Riemannian manifolds, for which the Riemannian metric admits various curvature invariants. Symplectic manifolds do however admit many global numerical invariants, and prominent among them are the so-called symplectic capacities.Symplectic capacities were introduced in 1990 by I. Ekeland and H. Hofer [19,20] (although the first capacity was in fact constructed by M. Gromov [40]). Since then, lots of new capacities have been defined [16,30,32,44,49,59,60,90,99] and they were further studied in [1,2,8,9,17,26,21,28,31,35,37,38,41,42,43,46,48,50,52,56,57,58,61,62,63,64,65,66,68,74,75,76,88,89,91,92,94,97,98]. Surveys on symplectic capacities are [45,50,55,69,97]. Different capacities are defined in different ways, and so relations between capacities often lead to surprising relations between different aspects of symplectic geometry and Hamiltonian dynamics. This is illustrated in § 2, where we discuss some examples of symplectic capacities and describe a few consequences of their existence. In § 3 we present an attempt to better understand the space of all symplectic capacities, and discuss some further general properties of symplectic capacities. In § 4, we describe several new relations between certain symplectic capacities on ellipsoids and polydiscs. Throughout the discussion we mention many open problems.As illustrated below, many of the quantitative aspects of symplectic geometry can be formulated in terms of symplectic capacities. Of course there are other numerical invariants of symplectic manifolds which could be included in *

show abstract

“…By (16) and the monotonicity of the f i we would find that c vol (E) ≤ c vol (F ). This is not true.…”

Section: Indeed If E(a)mentioning

confidence: 97%

“…Since then, lots of new capacities have been defined [16,30,32,44,49,59,60,90,99] and they were further studied in [1,2,8,9,17,26,21,28,31,35,37,38,41,42,43,46,48,50,52,56,57,58,61,62,63,64,65,66,68,74,75,76,88,89,91,92,94,97,98]. Surveys on symplectic capacities are [45,50,55,69,…”

mentioning

confidence: 99%

Quantitative symplectic geometry

Cieliebak¹,

Hofer²,

Latschev³

et al. 2007

Dynamics, Ergodic Theory and Geometry

Self Cite

107

View full text Add to dashboard Cite

show abstract

“…Together with the h-principle for Lagrangian immersions due to M. Gromov [26] and A. Lees [34], this implies that all Lagrangian tori in (R 4 , ω 0 ) are regular homotopic through Lagrangian immersions. In recent work K. Cieliebak and K. Mohnke [10] found the following extension of the former result to arbitrary dimensions: For any Lagrangian torus inside a symplectic vector spaces or a projective space, there exists a disk with boundary on the torus on which the Maslov class takes the value 2, and which is of positive symplectic area. In other words, the so-called Audin conjecture holds.…”

Section: 2mentioning

confidence: 85%

“…The splitting construction applied to the unit normal bundle of a Lagrangian submanifold has previously been used in [16,Theorem 1.7.5], [27], [14], [28], and [10], among others. Manifestly, it is an efficient tool for obtaining strong obstructions to Lagrangian embeddings inside uniruled symplectic manifolds.…”

Section: The Splitting Constructionmentioning

confidence: 99%

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

2016

View full text Add to dashboard Cite

Abstract. We show that, up to Lagrangian isotopy, there is a unique Lagrangian torus inside each of the following uniruled symplectic fourmanifolds: the symplectic vector space R 4 , the projective plane CP 2 , and the monotone S 2 × S 2 . The result is proven by studying pseudoholomorphic foliations while performing the splitting construction from symplectic field theory along the Lagrangian torus. A number of other related results are also shown. Notably, the nearby Lagrangian conjecture is established for T * T 2 , i.e. it is shown that every closed exact Lagrangian submanifold in this cotangent bundle is Hamiltonian isotopic to the zero-section.

show abstract

“…The existence of suitable hypersurfaces is a special geometric property of the symplectic manifold and the type of curves considered. It has been used in other restricted geometric situations, for example [CM2]. However, as mentioned in Remark 2.1.1, it is not yet clear whether it can be extended to higher genus curves as claimed in [Ge].…”

Section: Differentiability Issues In General Holomorphic Curve Modulimentioning

confidence: 99%

The fundamental class of smooth Kuranishi atlases with trivial isotropy

2017

View full text Add to dashboard Cite

Abstract. Kuranishi structures were introduced in the 1990s by Fukaya and Ono for the purpose of assigning a virtual cycle to moduli spaces of pseudoholomorphic curves that cannot be regularized by geometric methods. Their core idea was to build such a cycle by patching local finite dimensional reductions. The first sections of this paper discuss topological, algebraic and analytic challenges that arise in this program.We then develop a theory of Kuranishi atlases and cobordisms that transparently resolves these challenges, for simplicity concentrating on the case of trivial isotropy. In this case, we assign to a cobordism class of additive weak Kuranishi atlases both a virtual moduli cycle (VMC -a cobordism class of smooth manifolds) and a virtual fundamental class (VFC -a Cech homology class). We moreover show that such Kuranishi atlases exist on simple Gromov-Witten moduli spaces and develop the technical results in a manner that easily transfers to more general settings.

show abstract

Punctured holomorphic curves and Lagrangian embeddings

Cited by 55 publications

References 70 publications

Quantitative symplectic geometry

Quantitative symplectic geometry

Lagrangian isotopy of tori in $${S^2\times S^2}$$ S 2 × S 2 and $${{\mathbb{C}}P^2}$$ C P 2

The fundamental class of smooth Kuranishi atlases with trivial isotropy

Contact Info

Product

Resources

About