Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds

TL;DR

This paper demonstrates that spectral invariants on monotone symplectic manifolds descend and are continuous in the C^0-topology for Hamiltonian diffeomorphisms vanishing on open sets, with applications to Hofer geometry.

Contribution

It establishes C^0-rigidity and descent properties of spectral invariants on monotone symplectic manifolds, advancing understanding of Hofer geometry and metric properties.

Findings

Spectral invariants descend to Hamiltonian groups with compact support outside open sets.

Spectral invariants are continuous in the C^0-topology on these groups.

Unboundedness of Hofer diameter for certain Hamiltonian groups.

Abstract

We obtain estimates showing that on monotone symplectic manifolds (asymptotic) spectral invariants of Hamiltonians which vanish on a non-empty open set, U, descend to Ham_c(M\setminus U) from its universal cover. Furthermore, we show these invariants and are continuous with respect to the C^0-topology on Ham_c(M\setminus U). We apply these results to Hofer geometry and establish unboundedness of the Hofer diameter of $Ham_c(M\setminus U)$ for stably displaceable $U$. We also answer a question of F. Le Roux about $C^0$-continuity properties of the Hofer metric.