Spectral diameter of negatively monotone manifolds
Yuhan Sun
TL;DR
The paper addresses how negatively monotone symplectic manifolds constrain the size and geometry of their Hamiltonian diffeomorphism groups by establishing quasi-isometric embeddings of $(\mathbb{R}^N,|\cdot|_\infty)$ into $\text{Ham}(M,ω)$ via spectral invariants and heavy Lagrangians. The authors introduce a constructive approach that uses spectral invariants associated to functions supported near incompressible heavy Lagrangians, together with Entov–Polterovich heavy-set theory, to produce $N$-dimensional quasi-flats; they also exploit Lagrangian Floer cohomology to certify heaviness of incompressible Lagrangians and to identify super-heavy Lagrangian skeleta from Donaldson hypersurfaces. Key contributions include the main quasi-flat embedding result under negative monotonicity, heaviness of complements of Weinstein neighborhoods, and super-heaviness of the Biran–Giroux skeleton $L_Y$, with concrete implications for smooth hypersurfaces and product manifolds. Overall, the work advances understanding of the large-scale geometry and rigidity of the Hamiltonian diffeomorphism group in negatively monotone settings, providing tools to detect quasi-flats and to constrain displaceability via spectral norms.
Abstract
For a closed negatively monotone symplectic manifold, we construct quasi-isometric embeddings from the Euclidean spaces to its Hamiltonian diffeomorphism group, assuming it contains an incompressible heavy Lagrangian. We also show the super-heaviness of its skeleton with respect to a Donaldson hypersurface.