Hofer Energy and Link Preserving Diffeomorphisms in Higher Genus

TL;DR

The article addresses lower bounds for the Hofer energy of Hamiltonian diffeomorphisms preserving a premonotone Lagrangian link on a genus $g$ surface with boundary by transferring the problem to braid-type invariants on surface braid groups. It develops a Quantitative Heegaard–Floer framework that yields spectral invariants $c_{\underline L}(H)$ and a monotonicity theory that ensures nontrivial Floer homology, together with a Künneth formula that facilitates product decompositions. The central result constructs a family of Lipschitz-type homomorphisms $({\frak f}_{(v_1,v_2)})$ from the braid image to $\mathbb{R}$, providing sharp lower bounds $d_H(φ,ψ) \ge \tfrac{1}{2} |{\frak f}_{(v_1,v_2)}(b(φψ^{-1}))|$, and consequently $||φ|| \ge \tfrac{1}{2} \sup_{(v_1,v_2)} |{\frak f}_{(v_1,v_2)}(b(φ))|$ for $φ$ in the stabilizer of the Lagrangian link. The approach extends earlier disc results to higher genus, yields explicit values on generators, links to the Calabi morphism and asymptotic Hofer norms, and includes a Künneth-based alternative proof path. Overall, the work provides a robust braid-theoretic mechanism to quantify energy in Hamiltonian dynamics on complicated surfaces, with potential implications for symplectic topology and braid-type invariants on surfaces with boundary.

Abstract

Given a pre-monotone Lagrangian link, we obtain Hofer energy estimates for Hamiltonian diffeomorphisms preserving it. Such estimates depend on the braid type of the Hamiltonian diffeomorphism only, and the natural language to talk about this phenomenon is provided by a family of norms on braid groups for surfaces with boundary. This generalises the results obtained by the first author to higher genus surfaces with boundary.

Figures (8)

• Figure 1: The generators of the braid group $\mathcal{B}_{k, g, p}$. In clockwise order, starting from top-left: $\sigma_j$, $a_i$, $z_l$ and $b_i$ . We draw a fundamental domain of the surface, and all base points which are not endpoints of the green paths are to be thought of as constant paths.
• Figure 2: The generator $b_i^{-1}a_ib_i$
• Figure 3: The homotopy $\gamma(s,t)$
• Figure 4: The homotopy $\gamma(s, t)$ as seen on $\mathbb{C}$.
• Figure 5: The capping $\hat{z}_1$

Theorems & Definitions (55)

• Definition 1.1
• Remark 1.2
• Proposition 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Corollary 1.8
• Theorem 1.9
• Corollary 1.10