[Paper Review] Mini-max theory, spectral invariants and geometry of the Hamiltonian diffeomorphism group

Yong‐Geun Oh|ArXiv.org|Jun 10, 2002

Geometric and Algebraic Topology21 references15 citations

TL;DR

This paper introduces spectral invariants for Hamiltonian diffeomorphisms on arbitrary compact symplectic manifolds using mini-max theory over Novikov cycles in the universal cover of the contractible loop space. By constructing a continuous invariant norm via these invariants, the author establishes a new lower bound for the Hofer norm in terms of symplectic areas of pseudo-holomorphic curves and proves the semi-global $C^1$-flatness of the Hofer norm.

ABSTRACT

In this paper, we first develop a mini-max theory of the action functional over the semi-infinite cycles via the chain level Floer homology theory and construct spectral invariants of Hamiltonian diffeomorphisms on arbitrary compact symplectic manifold (M,omega). To each given time dependent Hamiltonian function H and quantum cohomology class 0 not equal a element of QH^*(M), we associate an invariant rho(H;a) which varies continuously over H in the C^0-topology. This is obtained as the mini-max value over the semi-infinite cycles whose homology class is `dual' to the given quantum cohomology class a on the covering space \widetilde Omega_0(M) of the contractible loop space Omega_0(M). We call them the Novikov cycles. We then use the spectral invariants to construct a new invariant norm on the Hamiltonian diffeomorphism group and a partial order on the set of time-dependent Hamiltonian functions of arbitrary compact symplectic manifolds. As some applications, we obtain a new lower bound of the Hofer norm of non-degenerate Hamiltonian diffeomorphisms in terms of the area of certain pseudo-holomorphic curves and prove the semi-global C^1-flatness of the Hofer norm.

Motivation & Objective

• To develop a mini-max theory for the action functional on semi-infinite cycles via chain-level Floer homology on arbitrary compact symplectic manifolds.
• To define spectral invariants $\rho(H; a)$ for time-dependent Hamiltonians $H$ and non-zero quantum cohomology classes $a \in QH^*(M)$, ensuring $C^0$-continuity.
• To construct an invariant norm on the Hamiltonian diffeomorphism group using spectral invariants, generalizing Hofer's geometry.
• To establish a lower bound for the Hofer norm in terms of symplectic areas of pseudo-holomorphic curves.
• To prove the semi-global $C^1$-flatness of the Hofer norm using spectral invariants and Hamiltonian fibrations.

Proposed method

• Construct spectral invariants $\rho(H; a)$ as mini-max values over semi-infinite cycles (Novikov cycles) dual to a given quantum cohomology class $a$ in the universal cover $\widetilde{\Omega}_0(M)$ of the contractible loop space.
• Use filtered Floer homology and the linking property of Novikov cycles to ensure finiteness and spectrality of the invariants.
• Define the invariant norm $\|\phi\|_\rho = \inf_H \rho(H; a)$ for $\phi \in \mathrm{Ham}(M,\omega)$, leveraging the triangle inequality derived via Hamiltonian fibrations.
• Apply the theory of Hamiltonian fibrations with fixed monodromy and pseudo-holomorphic sections to prove the triangle inequality and control $K$-area.
• Introduce bounded quantum cohomology $QH^*_{bdd}(M)$ to extend spectral invariants to a cohomologically well-behaved setting with continuous linear functionals.
• Use the $K$-area framework of Entov and connections on Hamiltonian fibrations to relate spectral invariants to Hofer-length-type quantities.

Experimental results

Research questions

• RQ1 How can spectral invariants be defined for Hamiltonian diffeomorphisms on non-exact symplectic manifolds where the action functional is multi-valued?
• RQ2 Can a continuous, invariant norm be constructed on $\mathrm{Ham}(M,\omega)$ using spectral invariants derived from Floer homology?
• RQ3 What is the relationship between the Hofer norm and the symplectic area of pseudo-holomorphic curves, and can this yield a lower bound?
• RQ4 How does the $C^1$-flatness of the Hofer norm manifest in the context of spectral invariants and Hamiltonian fibrations?
• RQ5 Can the spectral invariants be extended to bounded quantum cohomology classes, and what are the implications for the invariance and continuity of the norm?

Key findings

• The spectral invariant $\rho(H; a)$ is well-defined and continuous in the $C^0$-topology on the space of time-dependent Hamiltonians.
• A new invariant norm $\|\phi\|_\rho$ is constructed on $\mathrm{Ham}(M,\omega)$ using spectral invariants, satisfying the triangle inequality via Hamiltonian fibration techniques.
• A new lower bound for the Hofer norm of non-degenerate Hamiltonian diffeomorphisms is established in terms of the symplectic area of certain pseudo-holomorphic curves.
• The semi-global $C^1$-flatness of the Hofer norm is proven using the spectral invariants and the $K$-area framework of Hamiltonian fibrations.
• The spectral invariants extend to bounded quantum cohomology classes $QH^*_{bdd}(M)$, allowing a cohomologically robust formulation of the invariants.
• The construction generalizes Viterbo’s invariants and Hofer’s norm to arbitrary compact symplectic manifolds via Floer-theoretic and geometric methods.