Displacement-Energy Bounds for Poisson Brackets of Open Covers

Definitions and mathematical statement

Let $(M,\omega)$ be a closed symplectic manifold, where closed means compact and without boundary. The supplied record does not define the displacement-energy and Poisson-bracket conventions. The following gives an explicit candidate formulation using standard working conventions; their agreement with the cited report requires review.

For a smooth Hamiltonian $H:[0,1]\times M\to{\mathbb{R}}$, define its Hamiltonian vector field by $\iota_{X_{H_t}}\omega=-{\,\mathrm{d}} H_t$, and denote its flow by $\phi_H^t$. Put \[ \ell(H)=\int_0^1\left(\max_{x\in M}H_t(x)-\min_{x\in M}H_t(x)\right){\,\mathrm{d}} t. \] Under the working convention that displacement means $\phi_H^1(U)\cap U=\varnothing$, an open set $U\subset M$ is displaceable if such an $H$ exists, and its displacement energy is \[ e(U)=\inf\{\ell(H):\phi_H^1(U)\cap U=\varnothing\}. \] The record leaves unresolved whether displacement is instead required for the closure of $U$.

Let $\mathcal U=\{U_i\}_{i\in I}$ be a finite open cover of $M$. A smooth partition of unity subordinate to $\mathcal U$ is a family $f=(f_i)_{i\in I}$ with $f_i\in C^\infty(M,[0,1])$, $\operatorname{supp}(f_i)\subset U_i$, and $\sum_{i\in I}f_i=1$. For smooth real-valued functions $F,G$, use $\{F,G\}=\omega(X_F,X_G)$ and $\|F\|_\infty=\max_{x\in M}|F(x)|$. Define \[ \operatorname{pb}(f)=\sup_{a,b\in[-1,1]^I} \left\|\left\{\sum_{i\in I}a_i f_i,\sum_{j\in I}b_j f_j\right\}\right\|_\infty, \qquad \operatorname{pb}(\mathcal U)=\inf_{f\text{ subordinate to }\mathcal U}\operatorname{pb}(f), \] and, when all cover elements are displaceable, put \[ e(\mathcal U)=\max_{i\in I}e(U_i). \] The coefficient range and support convention in this definition of $\operatorname{pb}$ are explicit working conventions, not definitions supplied by the record.

The proposed statement is: for every closed symplectic manifold $(M,\omega)$, there exists a constant $C(M,\omega)>0$ such that, for every finite open cover $\mathcal U$ of $M$ by displaceable sets, \[ \operatorname{pb}(\mathcal U)\,e(\mathcal U)\geq C(M,\omega). \] The constant may depend on $(M,\omega)$, but must be independent of the cover, including its cardinality. No connectedness assumption or further restriction on dimension is supplied.

The supplied dated literature assessment labels the problem partially solved and reports a proof for closed symplectic surfaces, with the general closed-manifold statement remaining unresolved. These are supplied curation claims, not independently verified mathematical or literature conclusions.

Short English statement

For each fixed closed symplectic manifold, ask whether every finite cover by displaceable open sets has its Poisson-bracket invariant times its largest displacement energy bounded below by a positive constant depending only on the manifold and symplectic form. The definitions above specify provisional conventions requiring source review. The supplied curation reports the surface case as proved and the general case as unresolved.

Sources

• Local UnsolvedMath registry, supplied complete source record, permanent ID $30004745$, source problem code catalogue entry, title: "Displacement-Energy Bounds for Poisson Brackets of Open Covers." Collection: Oberwolfach Reports Open Problems; supplied proposed year: 2021. No individual proposer is specified.
• Supplied report citation: Dynamische Systeme (2022), Oberwolfach Reports, pp. 1735--1803, DOI 10.4171/owr/2021/33, https://doi.org/10.4171/owr/2021/33. The registry labels its statement corrected_verified and reports checking it against the official report; this draft does not independently verify that claim.
• Supplied literature reference, labeled "surface-case proof": https://doi.org/10.4171/CMH/487.
• Supplied literature reference, labeled "primary preprint": https://arxiv.org/abs/1705.02513.
• Literature assessment date: 2026-08-22. This is supplied dated curation derived from the OWR record, not an independently verified assessment. It attributes the dimension-free lower-bound problem to Polterovich, reports the closed symplectic surface case as proved, and labels the general closed-manifold statement unresolved. The supplied problem status is partially_solved. The record requests independent verification of time-sensitive status and cited claims.