[Paper Review] Maslov class rigidity for Lagrangian submanifolds via Hofer's geometry

Ely Kerman, Nil İpek Şirikçi|ArXiv.org|Aug 10, 2008

Geometric and Algebraic Topology28 references3 citations

TL;DR

This paper establishes new rigidity results for the Maslov class of Lagrangian submanifolds using Hofer's geometry, showing that displaceable Lagrangian products with factors admitting negative curvature metrics have bounded minimal Maslov numbers. By linking periodic orbits of Hamiltonian flows to geodesics and Conley-Zehnder indices, it proves that the minimal Maslov number $N_L$ satisfies $N_L \leq \frac{1}{2}\dim M + 2$, with a tighter bound $N_L \leq \frac{1}{2}\dim M + 1$ for orientable Lagrangians.

ABSTRACT

In this work, we establish new rigidity results for the Maslov class of Lagrangian submanifolds in large classes of closed and convex symplectic manifolds. Our main result establishes upper bounds for the minimal Maslov number of displaceable Lagrangian submanifolds which are product manifolds whose factors each admit a metric of negative sectional curvature. Such Lagrangian submanifolds exist in every symplectic manifold of dimension greater than six or equal to four. The proof utilizes the relations between closed geodesics on the Lagrangian, the periodic orbits of geometric Hamiltonian flows supported near the Lagrangian, and the length minimizing properties of these flows with respect to the negative Hofer length functional.

Motivation & Objective

• To establish rigidity results for the Maslov class of Lagrangian submanifolds without relying on Lagrangian Floer homology or holomorphic discs.
• To extend known rigidity phenomena beyond monotone or Floer-theoretic settings by using Hofer's length functional and Hamiltonian dynamics.
• To provide upper bounds on the minimal Maslov number for displaceable Lagrangian submanifolds that are products of manifolds with negative sectional curvature.
• To connect geometric properties of Lagrangians—such as closed geodesics—to spectral invariants of Hamiltonian flows via Conley-Zehnder indices.

Proposed method

• Utilizes Sikorav's result that autonomous Hamiltonian flows with displaceable support do not minimize Hofer length for all time.
• Applies a result from Hofer's geometry stating that non-length-minimizing flows possess contractible periodic orbits with spanning discs and specific Conley-Zehnder index bounds.
• Analyzes perturbed cogeodesic flows supported near displaceable Lagrangians to detect periodic orbits with Conley-Zehnder index $\frac{1}{2}\dim M$.
• Establishes a key identity linking the Conley-Zehnder index of such orbits to the Morse index of the corresponding perturbed geodesic and the Maslov index of the spanning disc.
• Uses this identity to derive constraints on the Maslov class by analyzing action values and length functionals.
• Applies the framework to product Lagrangians $L = L_1 \times L_2$, where $L_1$ is split hyperbolic and $L_2$ has no nonconstant contractible geodesics and is incompressible in $\pi_1$.

Experimental results

Research questions

• RQ1 What upper bounds can be established for the minimal Maslov number of displaceable Lagrangian submanifolds in symplectic manifolds of dimension $\geq 4$ or $>6$?
• RQ2 Can rigidity of the Maslov class be proven without using holomorphic discs or Lagrangian Floer homology?
• RQ3 How do the properties of closed geodesics on a Lagrangian relate to periodic orbits of Hamiltonian flows and their Conley-Zehnder indices?
• RQ4 What constraints does Hofer's geometry impose on the Maslov class when the ambient manifold is convex or closed and the Lagrangian is a product with negative curvature factors?
• RQ5 Can the minimal Maslov number be bounded in terms of the ambient dimension for orientable, easily displaceable Lagrangian products?

Key findings

• For any displaceable Lagrangian submanifold $L = L_1 \times L_2$ where $L_1$ admits a metric of negative sectional curvature and $L_2$ is incompressible with no nonconstant contractible geodesics, the minimal Maslov number satisfies $N_L \leq \frac{1}{2}\dim M + 2$.
• If $L$ is orientable, the bound tightens to $N_L \leq \frac{1}{2}\dim M + 1$.
• The proof detects nonconstant periodic orbits of Hamiltonian flows with Conley-Zehnder index exactly $\frac{1}{2}\dim M$, which are linked to perturbed geodesics on the Lagrangian.
• The minimal Maslov number is constrained by the action values of these orbits and the length properties of the Hofer functional.
• The results apply to all symplectic manifolds of dimension $\geq 4$ or $>6$ that are closed or convex and rational, and to Lagrangians that are products with split hyperbolic and incompressible factors.
• The framework avoids codimension one bubbling issues by bypassing holomorphic disc techniques and instead relying on geometric Hamiltonian dynamics and Hofer's functional.