Rigid subsets of symplectic manifolds

By Michael Entov, L. Polterovich. Compositio Mathematica, 2007

187 citations recorded by the source corpus.

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 others. We also find new examples of rigidity of intersections involving, in particular, specific fibers of moment maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following the works of P. Albers and P. Biran-O. Cornea) as well as certain, possibly singular, sets defined in terms of Poisson-commutative subalgebras of smooth functions. In addition, we get some geometric obstructions to semi-simplicity of the quantum homology of symplectic manifolds. The proofs are based on the Floer-theoretical machinery of partial symplectic quasi-states.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/388473ce0351b0524987ea4fa2f0e652cbcaec44

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Citations in the corpus

Rigidity and uniruling for Lagrangian submanifolds (2008, background): “Special thanks from both of us to Leonid Polterovich for interesting comments and his interest in this project from its early stages as well as for having pointed out a number of imprecisions in earlier versions of the paper.”
Microlocal condition for non-displaceablility (2008, background): “Same result has been proven in [ 3 ] using Hamiltonian Floer theory.”
Deformed Hamiltonian Floer theory, capacity estimates and Calabi quasimorphisms (2010, background): “M; !/ satisfies the stronger condition of being semisimple (for instance results of Entov and Polterovich [15] show that semisimplicity leads to stronger rigidity results in some contexts than does the property of having a field direct summand), we also define a notion of .”
Notes on monotone Lagrangian twist tori (2010, methodology): “Different constructions of an exotic torus in S(2) × S(2) were given by Biran– Cornea [5], Entov–Polterovich [14], Fukaya–Oh–Ohta–Ono [18].”
The wall-crossing formula and Lagrangian mutations (2017, background): “1] and [14], we recall the definition of monotone polytopes: they are the bases of toric fibrations on toric Fano manifolds, or monotone toric symplectic manifolds.”
Spectral invariants for monotone Lagrangians (2015, background): “ted coeﬃcients we have QH∗(L) 6= 0 (ibid.). It is known that the Lagrangian antidiagonal ∆ ⊂ Mis e−- superheavy. Since Lis disjoint from the antidiagonal, it follows that it is not e−-superheavy (see [EP09]), therefore Proposition 11 implies e− • [L] = 0. Therefore, since [M] = e+ +e−, we have e+ • ”
Symplectic quasi-states and semi-simplicity of quantum homology (2007, background): “theory to any non-zero quantum homology class a, any (timedependent) Hamiltonian Hon M[34] and any element φ∈ Ham](M,ω) (see 12 also [44, 31, 41, 32] for earlier versions of this theory). We refer to [15] for a detailed review. In what follows we assume that all the symplectic manifolds we deal with”
Displacing Lagrangian toric fibers via probes (2009, background): “cides with the special point u 0 and that HF(L u;˜) = 0 for all other u; cf. [8, Thm. 7.11]. One of the main motivating questions for the current study was raised by Entov and Polterovich, who ask in [6] whether the special ber L u 0 is stem, that is, whether all other bers are displaceable. The res”
Gauged Floer Theory Of Toric Moment Fibers (2010, methodology): “In a series of papers [21], [19] Fukaya-Oh-Ohta-Ono used this strategy to prove non-displaceability results for certain moment fibers of toric varieties, generalizing earlier work of Cho-Oh [12] and Entov-Polterovich [14].”
Bounds on spectral norms and barcodes (2018, background): “L];H) = c;and hence ˙([H]) = c;since the constant property is invariant under iterations. We note that for the same reason Lis superheavy with respect to the corresponding symplectic quasi-state (see [40, 65]). Finally, if H= Fh Fi;for a Hamiltonian F supported in a displaceable open subset UˆM; and”
Partial quasi-morphisms and quasi-states on cotangent bundles, and symplectic homogenization (2011, methodology): “Nis at most k, then max i<j k{fi,fj}kC0 ≥ C′ e(U)kK2 . The proof is a verbatim repetition of the one in [EPZ] and will be omitted. We now turn to non-displaceability. We refer the reader to [EP2], [EP3] for a treatment of the rigidity of subsets in closed symplectic manifolds. Following [EP3], we”
Quantum Structures for Lagrangian Submanifolds (2007, background): “Finally, let us mention that Entov and Polterovich [28] have also found restrictions on semi-simplicity of quantum homology related to Lagrangian submanifolds.”
Action and index spectra and periodic orbits in Hamiltonian dynamics (2008, methodology): “(Applications of action selectors extend far beyond the scope of our discussion here; see, e.g., [FS, EP1, EP2 , Gi1, G¨u, HZ, MS2, Sc1, Vi1] and references therein.) In this section, we mainly follow [Oh2], where the actions selectors are studied for Hamiltonians on closed, weakly monotone manifol”
Periodic Orbits of Twisted Geodesic Flows and The Weinstein-Moser Theorem (2007, methodology): “, [BG], [EP2], [RS], [SZ]) can be used in the proof of Theorem 1.”
Boundary depth in Floer theory and its applications to Hamiltonian dynamics and coisotropic submanifolds (2009, background): “) By now, there have been several applications of the spectral numbers; for a sampling of these, we refer the reader to [8], [13], [29], [43].”