Spectral invariants of distance functions (Q2828061)

The author constructs new examples of so-called superheavy subsets of symplectic manifolds. Introduced by \textit{M. Entov} and \textit{L. Polterovich} [Compos. Math. 145, No. 3, 773--826 (2009; Zbl 1230.53080)], the notion of a superheavy subset originates from spectral invariants of Hamiltonian functions: a closed subset \(X\) of a closed \(2n\)-dimensional symplectic manifold \(M\) is called superheavy with respect to an idempotent element \(\alpha\) of the \(2n\)-dimensional quantum cohomology of \(M\), if for every autonomous Hamiltonian on \(M\) the partial symplectic quasi-state of \(H\) with respect to \(\alpha\) is a lower bound for the maximum of \(H\) on \(X\). Here the partial symplectic quasistate of \(H\) is given by the asymptotic average for \(k \rightarrow \infty\) of a classical spectral invariant (detailed, e.g., in the works by \textit{M. Schwarz} [Pac. J. Math. 193, No. 2, 419--461 (2000; Zbl 1023.57020)] and \textit{Y.-G. Oh} [Prog. Math. 232, 525--570 (2005; Zbl 1084.53076)] of the multiple \(k \cdot H\). Applications stem from the non-displaceability of superheavy subsets by symplectic isotopies as well as from the fact that superheaviness implies estimates for partial symplectic quasistates.NEWLINENEWLINEThe main theorem of the paper (Theorem 1.1) establishes superheaviness of every subset \(X\) of a closed non-positively monotone symplectic maifold \(M\), so that \(X\) is given as the complement in \(M\) of a disjoint union of open subsets \(U_{j}\) that are symplectomorphic to convex subsets of Euclidean space \(\mathbb{R}^{2n}\) with its standard symplectic structure. Along with fundamental properties of superheavy subsets that are stated in the second section of the article, the proof relies on carefully established estimates for the actions and the Conley-Zehnder indices of the critical points of the function given (up to rescaling and mollifying) by the distance between a given point of \(M\) and the subset \(X\). From these an estimate for spectral invariants is concluded (Proposition 4.4), of which superheaviness is a corollary.NEWLINENEWLINEThe last section of the article contains some applications. Although from the point of view of displaceability questions, the main theorem does not seem to be very consequential due to purely topological constraints, following earlier work of \textit{S. Seyfaddini} [J. Mod. Dyn. 9, 51--66 (2015; Zbl 1325.53111)], the results allow to obtain lower estimates for the Poisson invariant bracket of finite covers of \(M\) in terms of the size of the subsets \(U_{j}\). As a second application, the author obtains a lower bound for the Hofer-Zehnder capacity of the complement of a special fiber in a monotone symplectic manifold equipped with a Hamiltonian torus action.