# Spherical Lagrangians via ball packings and symplectic cutting

@article{Borman2014SphericalLV, title={Spherical Lagrangians via ball packings and symplectic cutting}, author={Matthew Strom Borman and Tian-Jun Li and Weiwei Wu}, journal={Selecta Mathematica}, year={2014}, volume={20}, pages={261-283} }

In this paper, we prove the connectedness of symplectic ball packings in the complement of a spherical Lagrangian, $$S^{2}$$ or $$\mathbb{RP }^{2}$$, in symplectic manifolds that are rational or ruled. Via a symplectic cutting construction, this is a natural extension of McDuff’s connectedness of ball packings in other settings and this result has applications to several different questions: smooth knotting and unknottedness results for spherical Lagrangians, the transitivity of the action of…

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## References

SHOWING 1-10 OF 53 REFERENCES

Packing numbers of rational ruled 4-manifolds

- Mathematics
- 2011

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic 4-manifolds. We give explicit formulae for the packing numbers, the generalized Gromov…

Packing numbers of rational ruled four-manifolds

- Mathematics
- 2013

We completely solve the symplectic packing problem with equally sized balls for any rational, ruled, symplectic four-manifolds. We give explicit formulae for the packing numbers, the generalized…

Symplectic mapping class groups of some Stein and rational surfaces

- Mathematics
- 2009

In this paper we compute the homotopy groups of the symplectomorphism groups of the 3-, 4- and 5-point blow-ups of the projective plane (considered as monotone symplectic Del Pezzo surfaces). Along…

Isotopy of symplectic balls, Gromov's radius and the structure of ruled symplectic 4-manifolds

- Mathematics
- 1994

There are various criteria (Enriques, Castelnuovo, Mori) in algebraic geometry of complex surfaces that characterize rational or ruled surfaces up to birationality. Mori's criterion, in particular,…

Rigid subsets of symplectic manifolds

- MathematicsCompositio Mathematica
- 2009

Abstract We show that there is an hierarchy of intersection rigidity properties of sets in a closed symplectic manifold: some sets cannot be displaced by symplectomorphisms from more sets than the…

J-curves and the classification of rational and ruled symplectic 4-manifolds

- Mathematics
- 2009

The classification of symplectic structures on CP has followed a quite simple story: Gromov showed in [2] that the existence of an embedded sympectic 2sphere in the homology class [CP] of a line…

Symplectic genus, minimal genus and diffeomorphisms

- Mathematics
- 2001

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to…

On an exotic Lagrangian torus in $\mathbb{C}P^{2}$

- Physics, MathematicsCompositio Mathematica
- 2015

We find a non-displaceable Lagrangian torus fiber in a semi-toric system which is superheavy with respect to a certain symplectic quasi-state. The proof employs both 4-dimensional techniques and…

The structure of rational and ruled symplectic 4-manifolds

- Mathematics
- 1990

This paper investigates the structure of compact symplectic 4-manifolds (V, w) which contain a symplectically embedded copy C of S2 with nonnegative self-intersection number. Such a pair (V, C, w) is…

Lagrangian unknottedness in Stein surfaces

- Mathematics
- 2012

We show that the space of Lagrangian spheres inside the cotangent bundle of the 2-sphere, with its canonical symplectic structure, is contractible. We then discuss the phenomenon of Lagrangian…