Maximal symplectic packings in \({\mathbb {P}^2}\).

*(English)*Zbl 1133.53057Let \(B(r_i)\) be the standard \(2n\)-dimensional open ball of radius \(r_i\) in \(\mathbb{R}^{2n}\), endowed with the standard symplectic structure \(\omega_{st}\) of \(\mathbb{R}^{2n}\), \(\omega_{st}=\sum_{j=1}^n dx_j\wedge dx_{n+j}\). A smooth (resp., regular) maximal symplectic packing of a \(2n\)-dimensional symplectic manifold \((M^{2n},\omega)\) is a symplectic embedding \(\varphi :(B(r_1)\amalg\ldots\amalg B(r_k),\omega_{st})\to (M^{2n},\omega)\) of a disjoint union of \(k\) standard \(2n\)-dimensional open balls into \((M^{2n},\omega)\), where the radii are such that there exists no symplectic packing of \((M^{2n},\omega)\) by balls of radii \(r_1,\ldots,r_i+\varepsilon,\ldots,r_k\), and each \(\varphi_i:=\varphi| _{B(r_i)}\) extends to a smooth embedding of the closed ball (resp., each \(\varphi_i\) extends to a topological embedding of the closed ball which is smooth except a finite number of singular points). Consider \((\mathbb{C}P^2,\omega)\), where \(\omega\) is the standard Fubini-Study form of \(\mathbb{C}P^2\), conveniently normalized. Examples of smooth maximal symplectic packings of \(\mathbb{C}P^2\) were given by Y. Karshon [see the appendix of the paper by D. McDuff and L. Polterovich, Invent. Math. 115, 405–429 (1994; Zbl 0833.53028)] and L. Traynor [J. Differ. Geom. 41, 735–751 (1995; Zbl 0830.52011)].

The space of regular maximal symplectic packings by a fixed number of balls is naturally endowed with the Hausdorff topology for compact sets. In what follows, genericity is meant with respect to this topology: a property is generic if it is true for an open dense set. The author describes the intersections between the balls of maximal symplectic packings of \(\mathbb{C}P^2\). He uses an approach based on a strong link observed by G. P. Paternain, L. Polterovich and K. F. Siburg [Mosc. Math. J. 3, 593–619 (2003; Zbl 1048.53058)] or F. Laudenbach and J.-C. Sikorav [Int. Math Res. Not. 1994, No. 4, 161–168 (1994; Zbl 0812.53031)] between non-removable intersections in symplectic geometry and characteristic foliations to obtain his results:

1. Generically, the closed balls of a smooth maximal symplectic packing of \(\mathbb{C}P^2\) by two balls intersect precisely along one common Hopf circle of their boundary.

2. Generically, any two closed balls of a smooth maximal symplectic packing of \(\mathbb{C}P^2\) by three equal balls intersect precisely along one common Hopf circle of their boundary.

3. Generically, any two smallest balls of a smooth maximal symplectic packing of \(\mathbb{C}P^2\) by three non-equal balls do not intersect, while the intersection of the biggest ball with any of the others is exactly one common Hopf circle of their boundaries.

4. There exists no smooth maximal symplectic packing of \(\mathbb{C}P^2\) by more than three equal balls.

The previous results are generalized to the regular setting. The author’s analysis shows the existence of regular maximal symplectic packings of \(\mathbb{C}P^2\) by five equal balls. This existence result is sharpened when the singularities are simple enough.

The space of regular maximal symplectic packings by a fixed number of balls is naturally endowed with the Hausdorff topology for compact sets. In what follows, genericity is meant with respect to this topology: a property is generic if it is true for an open dense set. The author describes the intersections between the balls of maximal symplectic packings of \(\mathbb{C}P^2\). He uses an approach based on a strong link observed by G. P. Paternain, L. Polterovich and K. F. Siburg [Mosc. Math. J. 3, 593–619 (2003; Zbl 1048.53058)] or F. Laudenbach and J.-C. Sikorav [Int. Math Res. Not. 1994, No. 4, 161–168 (1994; Zbl 0812.53031)] between non-removable intersections in symplectic geometry and characteristic foliations to obtain his results:

1. Generically, the closed balls of a smooth maximal symplectic packing of \(\mathbb{C}P^2\) by two balls intersect precisely along one common Hopf circle of their boundary.

2. Generically, any two closed balls of a smooth maximal symplectic packing of \(\mathbb{C}P^2\) by three equal balls intersect precisely along one common Hopf circle of their boundary.

3. Generically, any two smallest balls of a smooth maximal symplectic packing of \(\mathbb{C}P^2\) by three non-equal balls do not intersect, while the intersection of the biggest ball with any of the others is exactly one common Hopf circle of their boundaries.

4. There exists no smooth maximal symplectic packing of \(\mathbb{C}P^2\) by more than three equal balls.

The previous results are generalized to the regular setting. The author’s analysis shows the existence of regular maximal symplectic packings of \(\mathbb{C}P^2\) by five equal balls. This existence result is sharpened when the singularities are simple enough.

Reviewer: Mircea Craioveanu (Timişoara)

##### MSC:

53D35 | Global theory of symplectic and contact manifolds |

53D05 | Symplectic manifolds (general theory) |

57R40 | Embeddings in differential topology |

57R17 | Symplectic and contact topology in high or arbitrary dimension |