[Paper Review] Length minimizing property, Conley-Zehnder index and $C^1$-perturbations of Hamiltonian functions

Yong‐Geun Oh|ArXiv.org|Feb 10, 2004

Geometric and Algebraic Topology7 references3 citations

TL;DR

This paper establishes conditions under which quasi-autonomous Hamiltonian paths are length-minimizing in their homotopy class on very strongly semi-positive symplectic manifolds, using $C^1$-perturbations and spectral invariants from chain-level Floer homology. The key result identifies a class of positively $μ$-undertwisted Hamiltonians—admitting non-constant periodic orbits—whose paths minimize Hofer length under fixed endpoints, linking the Minimality Conjecture to $C^1$-perturbation problems.

ABSTRACT

The main purpose of this paper is to study the length minimizing property of Hamiltonian paths on closed symplectic manifolds $(M,ω)$ such that there are no spherical homology class $A \in H_2(M)$ with $$ ω(A) > 0 \quad ext{and} \quad -n \leq c_1(A) < 0, $$ which we call {\it very strongly semi-positive}. We introduce the notion of {\it positively $μ$-undertwisted} Hamiltonian paths and prove that any positively undertwisted quasi-autonomous Hamiltonian path is length minimizing in its homotopy class as long as it has a fixed maximum and a fixed minimum point that are generically under-twisted. This class of Hamiltonian can have non-constant large periodic orbits. The proof uses the chain level Floer theory, spectral invariants of Hamiltonian diffeomorphisms and the argument involving the thick and thin decomposition of Floer's moduli space of perturbed Cauchy-Riemann equation. And then based on this theorem and some closedness of length minimizing property, we relate the Minimality Conjecture on the very strongly semi-positive symplectic manifolds to a $C^1$-perturbation problem of Hamiltonian functions on general symplectic manifolds, which we also formulate here.

Motivation & Objective

• To investigate the length-minimizing property of Hamiltonian paths on closed symplectic manifolds without requiring the absence of non-constant periodic orbits.
• To formulate and prove a minimality condition for quasi-autonomous Hamiltonian paths under topological constraints on Conley-Zehnder indices.
• To relate the Minimality Conjecture on very strongly semi-positive manifolds to a $C^1$-perturbation problem of Hamiltonian functions on general symplectic manifolds.
• To establish a framework using chain-level Floer theory and spectral invariants to analyze path minimality in the presence of nontrivial periodic orbits.

Proposed method

• Introduces the notion of positively $μ$-undertwisted Hamiltonian paths, generalizing slow autonomous Hamiltonians.
• Applies chain-level Floer homology and spectral invariants to track changes in Conley-Zehnder indices under perturbations.
• Uses the thick and thin decomposition of Floer's moduli space of perturbed Cauchy-Riemann solutions to control compactness and transversality.
• Employs trivialization-dependent Conley-Zehnder index formulas involving winding numbers of transition matrices to compute index differences.
• Derives the identity $\mu_H([z,w]) = \mu_H([z,w^\prime]) + 2c_1([w\#\overline{w}^\prime])$ relating index shifts to Chern numbers of sphere bundles.
• Establishes closedness of the length-minimizing property under $C^1$-perturbations, enabling reduction of the Minimality Conjecture to a perturbation problem.

Experimental results

Research questions

• RQ1 Under what conditions is a quasi-autonomous Hamiltonian path length-minimizing in its homotopy class with fixed endpoints?
• RQ2 How do Conley-Zehnder indices and spectral invariants constrain the minimality of Hamiltonian paths in the presence of non-constant periodic orbits?
• RQ3 Can the Minimality Conjecture on very strongly semi-positive manifolds be reduced to a $C^1$-perturbation problem of Hamiltonian functions?
• RQ4 What role does the $C^1$-topology play in preserving or breaking the length-minimizing property of Hamiltonian paths?
• RQ5 How does the $\mu$-undertwisted condition interact with the spectral invariants and Floer-theoretic constructions?

Key findings

• Any positively $μ$-undertwisted quasi-autonomous Hamiltonian path with fixed maximum and minimum points that are generically under-twisted is length-minimizing in its homotopy class.
• The paper proves Theorem A' and Theorem B, establishing the length-minimizing property under the very strongly semi-positive condition on the symplectic manifold.
• The Conley-Zehnder index satisfies the identity $\mu_H([z,w]) = \mu_H([z,w^\prime]) + 2c_1([w\#\overline{w}^\prime])$, linking index differences to Chern numbers of sphere bundles.
• The class of very strongly semi-positive manifolds includes Fano manifolds, monotone symplectic manifolds, and weakly exact symplectic manifolds.
• The Minimality Conjecture is reduced to a $C^1$-perturbation problem of Hamiltonian functions on general symplectic manifolds, under the assumption of very strong semi-positivity.
• The proof relies on elementary Floer theory without virtual moduli cycles, valid under the very strongly semi-positive condition.