[Paper Review] Descent and C^0-rigidity of spectral invariants on monotone symplectic manifolds

Sobhan Seyfaddini|arXiv (Cornell University)|Jul 10, 2012

Geometric and Algebraic Topology13 references5 citations

TL;DR

This paper establishes $C^0$-rigidity estimates for spectral invariants on monotone symplectic manifolds, showing that the difference between spectral invariants of two compactly supported Hamiltonians depends linearly on the $C^0$-distance between their time-1 maps, plus a topological correction term involving the monotonicity constant $\lambda$. The key result is that spectral invariants descend from the universal cover of the compactly supported Hamiltonian group to the group itself when $\lambda > 0$, with applications to Hofer geometry and $C^0$-continuity of the Hofer metric.

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.

Motivation & Objective

• To establish $C^0$-rigidity estimates for spectral invariants that depend only on the $C^0$-distance between time-1 Hamiltonian diffeomorphisms, not the entire paths.
• To resolve the question of whether spectral invariants descend from the universal cover of $Ham_c(M\setminus U)$ to $Ham_c(M\setminus U)$ on monotone symplectic manifolds.
• To apply the results to Hofer geometry, proving unboundedness of the Hofer diameter of $Ham_c(M\setminus U)$ for stably displaceable open sets $U$.
• To answer a question of F. Le Roux on the $C^0$-continuity properties of the Hofer metric by showing that the set $\mathcal{E}_A(M\setminus U)$ has non-empty $C^0$-interior for any $A>0$.

Proposed method

• Uses spectral invariants defined via Hamiltonian Floer homology, constructed by Schwarz and Oh, on closed, monotone symplectic manifolds with $[\omega]|_{\pi_2} = \lambda c_1|_{\pi_2}$.
• Applies a $\delta$-regularization procedure to perturb Hamiltonians supported away from a non-empty open set $U$, ensuring control over $C^0$-distances.
• Employs a $\epsilon$-shift argument in the symplectic manifold $M \setminus U$ to relate the $C^0$-distance of time-1 maps to spectral invariants.
• Establishes a key inequality: $|c(a,G) - c(a,H)| \leq C \cdot d_{C^0}(\phi^1_G, \phi^1_H) + n \max(0, \lambda)$, where $C$ depends on $U$.
• Extends the results to compact symplectic manifolds with boundary by embedding them into an open symplectic manifold via a cylindrical end construction.
• Applies the theory of admissible Hamiltonians on the extended manifold $\hat{X} = X \cup_{\partial X} \partial X \times [0,\infty)$ to define spectral invariants consistently.

Experimental results

Research questions

• RQ1 Does the $C^0$-distance between time-1 Hamiltonian diffeomorphisms control the difference in their spectral invariants on monotone symplectic manifolds, independent of the path?
• RQ2 Under what conditions do spectral invariants on $\widetilde{Ham_c}(M\setminus U)$ descend to $Ham_c(M\setminus U)$?
• RQ3 Is the Hofer diameter of $Ham_c(M\setminus U)$ unbounded when $U$ is stably displaceable?
• RQ4 Does the set $\mathcal{E}_A(M\setminus U)$ have non-empty $C^0$-interior for every $A>0$, resolving a question of F. Le Roux?

Key findings

• The paper establishes a $C^0$-rigidity estimate: $|c(a,G) - c(a,H)| \leq C \cdot d_{C^0}(\phi^1_G, \phi^1_H) + n \max(0, \lambda)$, where $C$ depends on the open set $U$, for Hamiltonians supported in $M \setminus U$.
• Spectral invariants descend from $\widetilde{Ham_c}(M\setminus U)$ to $Ham_c(M\setminus U)$ on monotone symplectic manifolds when $\lambda > 0$, due to the topological correction term.
• The Hofer diameter of $Ham_c(M\setminus U)$ is unbounded when $U$ is stably displaceable, as spectral invariants can be made arbitrarily large.
• The set $\mathcal{E}_A(M\setminus U)$ has non-empty $C^0$-interior for every $A > 0$, affirmatively answering a question of F. Le Roux on $C^0$-continuity of the Hofer metric.
• The results extend to compact symplectic manifolds with boundary under monotonicity, where spectral invariants satisfy $|c(a,G) - c(a,H)| \leq C \cdot d_{C^0}(\phi^1_G, \phi^1_H) + n\lambda$.
• The construction of spectral invariants via admissible Hamiltonians on $\hat{X} = X \cup_{\partial X} \partial X \times [0,\infty)$ ensures consistency and enables the extension of the main results to manifolds with boundary.