Coisotropic rigidity and C0-symplectic geometry

Abstract

Abstract. We prove that symplectic homeomorphisms, in the sense of the celebrated Gromov-Eliashberg Theorem, preserve coisotropic submanifolds and their characteristic foliations. This result generalizes the Gromov-Eliashberg Theorem and demonstrates that previous rigidity results (on Lagrangians by Laudenbach-Sikorav, and on characteristics of hypersurfaces by Opshtein) are manifestations of a single rigidity phenomenon. To prove the above, we establish a C 0 -dynamical property of coisotropic submanifolds which generalizes a foundational theorem in C 0 -Hamiltonian dynamics: Uniqueness of generators for continuous analogs of Hamiltonian flows.

Observations on the Hofer distance between closed subsets

“…The estimates in [BM13], [HLS13] that motivated this section were lower bounds for the righthand side in the above corollary; we thus see that any of the numerous methods for estimating e(A, U) in fact yields a similar estimate for this right-hand side.…”

“…In this section we give a simple explanation for these results which give estimates for (1) instead of only for the Hofer norm: they do not, as might first appear, represent some new mysterious action-at-a-distance phenomenon in symplectic topology; rather, by means of elementary considerations about the relationships between Hamiltonians and their time-one maps we will see that the sorts of Hofer norm bounds described above immediately imply identical bounds on the quantity (1). 2 In particular all of the bounds described in the first paragraph of this section can be combined with Theorem 1.3 below to yield bounds on (1) in the style of [BM13], [HLS13].…”

Reduction of symplectic homeomorphisms

“…First, notice that by definition the restrictions to C of f •φ and g coincide. Since g is constant on the characteristic leaves of C, its Hamiltonian flow φ t g preserves C. Thus H = (f •φ−g)•φ t g vanishes on C for all t. By [9,Theorem 3], the flow of the continuous Hamiltonian 2 H follows the characteristic leaves of C. On the other hand we know that this flow is given by the formula φ t H = (φ t g ) −1 φ −1 φ t f φ. This isotopy descends to the reduction R where it 1 This is always locally true.…”

“…We begin by first supposing that the reduction φ R is smooth. It turns out that this scenario can be resolved rather easily using a result of [9]. Proposition 2.…”

Spectral invariants in Lagrangian Floer homology of open subset

“…In [20], Leclercq constructed spectral invariants for Lagrangian Floer theory in case when L is a closed submanifold of a compact (or convex in infinity) symplectic manifold P and ω| π2(P,L) = 0, µ| π2(P,L) = 0, where µ is Maslov index. Symplectic invariants were further investigated by Eliashberg and Polterovich [11], Polterovich and Rosen [32], Oh [30], Humilière, Leclercq and Seyfaddini [13], by Monzner, Vichery and Zapolsky [26], Lanzat [18] and also in [9], [21,22,23,24].…”