Hofer's metrics and boundary depth

By Michael Usher. 2011

63 citations recorded by the source corpus.

We show that if (M,\omega) is a closed symplectic manifold which admits a nontrivial Hamiltonian vector field all of whose contractible closed orbits are constant, then Hofer's metric on the group of Hamiltonian diffeomorphisms of (M,\omega) has infinite diameter, and indeed admits infinite-dimensional quasi-isometrically embedded normed vector spaces. A similar conclusion applies to Hofer's metric on various spaces of Lagrangian submanifolds, including those Hamiltonian-isotopic to the diagonal in M x M when M satisfies the above dynamical condition. To prove this, we use the properties of a Floer-theoretic quantity called the boundary depth, which measures the nontriviality of the boundary operator on the Floer complex in a way that encodes robust symplectic-topological information.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a3a16723ab7356ecb73b063a130f5459aa1ec1f8

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Citations in the corpus

Persistent homology and Floer-Novikov theory (2015, methodology): “ each of the conditions (i)-(iv) in the deﬁnition of a singular value decomposition. 4. Boundary depth and torsion exponents via singular value decompositions The boundary depth as deﬁned in [U11] or [U13] is a numerical invariant of a ﬁltered chain complex that, in the case of the Hamiltonian and L”
Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules (2014, unknown): “Alternatively, taking ς to be the maximal length of a finite bar in B, we recover the boundary depth of φ as defined by Usher in [48].”
Persistence modules with operators in Morse and Floer theory (2017, methodology): “Assuming M is α-toroidally monotone, to every element f̃ ∈ H̃am(M) of the universal cover of Ham(M), that is non-degenerate in class α, we associate a Floer persistence module HF t ∗(f̃)α with parameter t (see [47, 55]).”
Bounds on spectral norms and barcodes (2018, methodology): “Recently, persistence modules found applications in symplectic topology, see for example [6, 42, 93, 94, 108, 116, 120], with precursors in [10, 32, 46, 113, 114].”