Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0212337

Spectral invariants and length minimizing property of Hamiltonian paths

Yong-Geun Oh
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper we provide a criterion for the quasi-autonomous Hamiltonian path (``Hofer's geodesic'') on arbitrary closed symplectic manifolds $(M,ω)$ to be length minimizing in its homotopy class in terms of the spectral invariants $ρ(G;1)$ that the author has recently constructed (math.SG/0206092). As an application, we prove that any autonomous Hamiltonian path on arbitrary closed symplectic manifolds is length minimizing in {\it its homotopy class} with fixed ends, when it has no contractible periodic orbits {\it of period one}, has a maximum and a minimum point which are generically under-twisted and all of its critical points are nondegenerate in the Floer theoretic sense. This is a sequel to the papers math.SG/0104243 and math.SG/0206092.

Explore related subjects

math.SG

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yong-Geun Oh. 2003-12-09. Spectral invariants and length minimizing property of Hamiltonian paths. https://arxiv.org/abs/math/0212337

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Multiplicity Bounds for Contractible Closed Reeb Orbits on Prequantization Bundles

We study the multiplicity of contractible closed Reeb orbits on prequantization circle bundles $(M^{2n+1},ξ)$ over closed symplectic manifolds $(B^{2n},ω)$. For even $n$, under suitable geometric hypotheses, we prove that every non-degenerate contact form supporting $ξ$ whose contractible closed orbits have nonzero mean index and whose good contractible closed orbits have nonzero Conley--Zehnder index carries at least $\frac{1}{2}χ(B)+\frac{1}{2}\dim H_n(B;\mathbb{Q})+1$ geometrically distinct contractible closed Reeb orbits. In dimension five, under the corresponding geometric hypotheses and assuming $H_1(B;\mathbb{Q})=0$, we prove the conjectural lower bound $\dim H_*(B;\mathbb{Q})$ for every non-degenerate supporting contact form whose simple contractible closed orbits have nonzero Conley--Zehnder index.

math.SG↗
arXiv2026

Categorical and Floer-theoretic entropies: comparison and counting conjugacy classes in symplectic mapping class groups

Let $W$ be a Weinstein manifold and let $f: W \to W$ be a compactly supported symplectic automorphism. Using Floer theories, one can define two different entropies of $f$, called "Floer-theoretic" entropy and "categorical" entropy. The current article compares these two entropies and show that the former bounds the latter from below. As an application of the comparison, using that these entropies are invariants up to the conjugate relation, we count conjugacy classes in symplectic mapping classes. Especially, we show an exponential abundance of conjugacy classes that are smoothly isotopic to the identity, when $W$ is a Weinstein manifold of dimension $4n \geq 6$, satisfying a local condition.

math.SG↗
arXiv2026

Mapping-space Gromov--Witten classes and the Fano variety of lines

Let $Y$ be a smooth cubic 3-fold. We show that $S^2 \times Y$ admits an infinite sequence of cohomologous symplectic forms which are all deformation equivalent but pairwise non-isotopic; McDuff showed the corresponding result on $(S^2)^2 \times T^2$, but these appear to be the first simply-connected and monotone examples. The results are obtained from McDuff's bordism-flavoured Gromov--Witten invariant, re-interpreted cohomologically in terms of an evaluation map valued in the space of maps from the 2-sphere to a symplectic target. The application then relies on classical properties of the Fano space of lines on $Y$. We also discuss higher-dimensional generalisations.

math.SG↗