Persistence-like distance on Tamarkin’s category and symplectic displacement energy. (English) Zbl 1473.18012
The displacement energy \(e(A,B)\), for two compact subsets \(A\) and \(B\) of the cotangent bundle \(T^*M\), measures the minimal energy of Hamiltonian isotopies necessary to displace the subset \(B\) in order to become disjoint with \(A\). The authors introduce a pseudo-distance on Tamarkin’s category \(\mathcal{D}(M)\), inspired from the work of [M. Kashiwara and P. Schapira, J. Appl. Comput. Topol. 2, No. 1–2, 83–113 (2018; Zbl 1423.55013)] on interleaving distance for persistence modules in the context of sheaves, and give in terms of it a lower bound for the displacement energy. As a corollary, Tamarkin’s separation theorem [D. Tamarkin, Springer Proc. Math. Stat. 269, 99–223 (2018; Zbl 1416.35019)] follows from this lower bound.
The very clearly and well written article carefully proves all results or gives references for known ones. It finishes with several applications and concrete calculi.
The very clearly and well written article carefully proves all results or gives references for known ones. It finishes with several applications and concrete calculi.
Reviewer: Thomas Krantz (Paris)
MSC:
18F20 | Presheaves and sheaves, stacks, descent conditions (category-theoretic aspects) |
35A27 | Microlocal methods and methods of sheaf theory and homological algebra applied to PDEs |
53D05 | Symplectic manifolds (general theory) |
32F45 | Invariant metrics and pseudodistances in several complex variables |