[Paper Review] Heavy subsets and non-contractible trajectories

Morimichi Kawasaki|arXiv (Cornell University)|Jun 6, 2016

Geometric and Algebraic Topology20 references3 citations

TL;DR

This paper introduces a new approach to estimating Biran-Polterovich-Salamon's relative symplectic capacity using Oh-Schwarz spectral invariants from Hamiltonian Floer theory on contractible trajectories, rather than non-contractible ones. It proves that for $n$-stably non-displaceable subsets $X$ in a closed symplectic manifold, the capacity $C(M,X,R;e)$ equals $\sum_{i=1}^n R_i \cdot |e_i|$, supporting a conjecture on stable non-displaceability and capacity finiteness.

ABSTRACT

Entov and Polterovich defined heaviness for closed subsets of a symplectic manifold by using the Hamiltonian Floer theory on contractible trajectories. Heavy subsets are known to be non-displaceable. In the present paper, we define a relative symplectic capacity $C(M,X,R;e)$ for a symplectic manifold $(M,ω)$ and its subset $X$ which measures the existence of non-contractible trajectories of Hamiltonian isotopies on the product with annulus. We prove that $C(M,X,R;e)$ is finite if $(M,ω)$ is monotone and $X$ is a heavy subset. We also prove that $C(M,X,R;e)$ is infinite if $X$ is a displaceable compact subset.

Motivation & Objective

• To re-estimate Biran-Polterovich-Salamon’s relative symplectic capacity using spectral invariants from contractible Hamiltonian Floer trajectories instead of non-contractible ones.
• To investigate whether Biran-Polterovich-Salamon’s capacity is finite for non-displaceable subsets, particularly in the context of stable non-displaceability.
• To support Conjecture 1.3, which posits that $C(M,X,R;e) = \sum R_i |e_i|$ for $n$-stably non-displaceable compact subsets $X$.
• To define and analyze a new capacity $C^P$ based on invariant measures of time-independent Hamiltonian flows, providing a lower bound for $C(M,X,R;e)$.

Proposed method

• Uses Oh-Schwarz spectral invariants derived from Hamiltonian Floer homology on contractible trajectories to estimate the Biran-Polterovich-Salamon capacity.
• Introduces a relative capacity $C(M,X,R;e)$ defined on $M \times I_R^n \times T^n$ with symplectic form $\operatorname{pr}_1^*\omega + \operatorname{pr}_2^*\omega_0$, where $X$ is a compact subset of a closed symplectic manifold $M$.
• Defines $n$-stably non-displaceable subsets as those for which $X \times T^n$ is non-displaceable in $M \times T^*T^n$, generalizing the notion of stable displaceability.
• Constructs a symplectic isotopy $\{\psi^t\}$ on $M \times T_{2R}^n \times T^n$ with flux matching a given cohomology class, enabling application of Theorem 9.4 on invariant measures.
• Introduces a capacity $C^P$ using closed 1-forms and invariant measures of Hamiltonian flows, satisfying $C^P \leq C$, and proves $C^P = \sum R_i |e_i|$ under the same assumptions.
• Employs a cutoff function $\rho$ on $I_R^n$ with controlled derivatives to construct a Hamiltonian $H(x,p,q) = \rho(p)$, ensuring $\int \lambda(X_H) \, d\mu \leq \lambda((0_M,e))$.

Experimental results

Research questions

• RQ1 Can spectral invariants from contractible trajectories be used to estimate the Biran-Polterovich-Salamon capacity, which traditionally relies on non-contractible trajectories?
• RQ2 Is the capacity $C(M,X,R;e)$ finite for $n$-stably non-displaceable subsets $X$?
• RQ3 Does $C(M,X,R;e)$ equal $\sum_{i=1}^n R_i \cdot |e_i|$ for $n$-stably non-displaceable $X$, as conjectured?
• RQ4 Can the new capacity $C^P$, based on invariant measures, provide a lower bound for $C(M,X,R;e)$, and does it achieve equality with $\sum R_i |e_i|$?
• RQ5 What is the role of stable non-displaceability in ensuring finiteness and exactness of the capacity?

Key findings

• The paper proves that for any $n$-stably non-displaceable compact subset $X$ of a closed symplectic manifold $M$, the capacity $C(M,X,R;e)$ satisfies $C(M,X,R;e) = \sum_{i=1}^n R_i \cdot |e_i|$.
• The capacity $C^P(M,X,R;e)$, defined via invariant measures of Hamiltonian flows, satisfies $C^P(M,X,R;e) = \sum_{i=1}^n R_i \cdot |e_i|$ under the same assumptions.
• The construction of a Hamiltonian $H(x,p,q) = \rho(p)$ with $\rho$ compactly supported and $|\partial\rho/\partial p_i| < |e_i|$ ensures that $\int \lambda(X_H) \, d\mu \leq \lambda((0_M,e))$, which is key to the lower bound.
• The proof uses a symplectic isotopy $\{\psi^t\}$ with flux matching $K\mathbf{l}^*$, where $K$ is chosen so that $(Ka_1,\dots,Ka_n) \in S_R$, enabling application of Theorem 9.4.
• The result supports Conjecture 1.3, showing that $C(M,X,R;e)$ is finite and exactly equal to $\sum R_i |e_i|$ for $n$-stably non-displaceable sets.
• An example is provided where $C(M,X,R;e) = +\infty$ even though $X$ is non-displaceable, indicating that stable non-displaceability is a necessary condition for finiteness.