Chain level Floer theory and Hofer's geometry of the Hamiltonian diffeomorphism group

By Yong-Geun Oh. 2001

91 citations recorded by the source corpus.

In this paper we first apply the chain level Floer theory to the study of Hofer's geometry of Hamiltonian diffeomorphism group in the cases without quantum contribution: we prove that any quasi-autonomous Hamiltonian path on weakly exact symplectic manifolds or any autonomous Hamiltonian path on arbitrary symplectic manifolds is length minimizing in its homotopy class with fixed ends, as long as it has a fixed maximum and a fixed minimum which are not over-twisted and all of its contractible periodic orbits of period less than one are sufficiently $C^1$-small. Next we give a construction of new invariant norm of Viterbo's type on the Hamiltonian diffeomorphism group of arbitrary compact symplectic manifolds.

Subfields: Biomaterials, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/154e8b356bedc3276b55fc50ea67a25989dba007

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Biomaterials

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Autonomous Hamiltonian flows, Hofer’s geometry and persistence modules (2014, background): “By [36] Specα(H) ⊂ R is a measure-zero subset, and hence has empty interior.”
Calabi quasi-morphisms for some non-monotone symplectic manifolds (2005, background): “c(H), as Spec(H) := n A H(x;u) 2R j[x;u] 2Pf H o : Algebraic & Geometric Topology 6 (2006) 418 Yaron Ostrover Recall that the action spectrum is either a discrete or a countable dense subset of R [26]. We now turn to give the deﬁnition of the ﬁltered Floer homology group. For a generic H 2Hand 2”
Spectral numbers in Floer theories (2007, methodology): “As mentioned earlier, Hamiltonian Floer homology is the Floer theory for which the spectral numbers have been most heavily developed, beginning with Schwarz’s work [13] and continuing with papers by Oh such as [ 8 ], [9] (in turn, Schwarz’s work was motivated in part by earlier work of Viterbo and O”
The Conley Conjecture (2006, unknown): “, [Gin07], [GG04], [Hin09], [HZ90], [HZ94], [Ker05], [KL03], [LM95], [MS01], [Oh02].”
Commutator length of symplectomorphisms (2001, background): “Denote by Spec(H) the set of values of AH on the elements of P(H). Proposition 2.3.1 ([22], [ 33 ], [47]).”
From symplectic measurements to the Mahler conjecture (2013, background): “Since Gromov’s work, several other symplectic capacities were constructed, such as the Hofer–Zehnder capacity [15], Ekeland–Hofer capacities [6], the displacement energy [14], spectral capacities [7, 23, 28], and Hutchings’ embedded contact homology capacities [16].”
Quantitative symplectic geometry (2005, unknown): “Using the Floer homology of (M,ω) filtered by the action functional, an action selector can be constructed for many (and conceivably for all) symplectic manifolds (M,ω), [32, 78, 79, 80, 88].”
Bounds on spectral norms and barcodes (2018, background): “These are known [83] to be closed nowhere dense subsets of R.”
Construction of spectral invariants of Hamiltonian paths on closed symplectic manifolds (2004, methodology): “The main purpose of the present paper is to further develop the chain level Floer theory introduced in [ Oh4 ] and to carry out construction of spectral invariants for arbitrary quantum cohomology classes.”
Geometric variants of the Hofer norm (2001, background): “∗Partially supported by NSF grant DMS0072512”
THE SHARP ENERGY-CAPACITY INEQUALITY (2008, background): “By Lemma 2.2 of [ 18 ], ( H) has Lebesgue measure zero.”
LECTURES ON FLOER THEORY AND SPECTRAL INVARIANTS OF HAMILTONIAN FLOWS (2006, methodology): “And the same result with the condition (1) is replaced by the one in which the phrase “of period one” replaced by “of period less than equal to one” was proven in [ Oh5 ] earlier.”