Observations on the Hofer distance between closed subsets

We prove the elementary but surprising fact that the Hofer distance between two closed subsets of a symplectic manifold can be expressed in terms of the restrictions of Hamiltonians to one of the subsets; this helps explain certain energy-capacity inequalities that appeared recently in [BM13] and [HLS13]. We also build on [U14] to obtain new vanishing results for the Hofer distance between subsets, applicable for instance to singular analytic subvarieties of Kahler manifolds. This note uses rather elementary arguments to deduce some results about the Hofer distance between closed subsets, defined as the infimal Hofer norm of a Hamiltonian diffeomorphism that maps one subset to the other. In the first section we give an alternative formula (Theorem 1.3) for this distance, which helps explain some seemingly-unexpectedly-strong versions of energycapacity inequalities that appeared recently in [BM13] and [HLS13], and indeed shows that all energy-capacity inequalities can be expressed in a similar strengthened form. The second section contains new results about the rigid locus defined in [U14], in particular connecting it to the Poisson bracket in Corollary 2.3, and uses these to expand the class of subsets on whose orbits the Hofer distance is known to vanish identically. Specifically this vanishing is established for all non-Lagrangian, half-dimensional submanifolds (Corollary 2.7) and all analytic subvarieties (including singular ones) in Kahler manifolds (Theorem 2.15). 1. RESTRICTING THE HAMILTONIAN There exists a rich history of results in symplectic topology asserting that, in order for a Hamiltonian diffeomorphism φ of a symplectic manifold (M ,ω) to behave in a certain way with respect to a subset A of M , the Hofer norm ‖φ‖H of φ must exceed some positive lower bound. Here ‖φ‖H is by definition the infimal value of ∫ 1 0 maxM H(t, ·)−minM H(t, ·) d t among smooth compactly supported functions H : [0,1] × M → R having time-one map φ H equal to φ. Indeed, the original proofs that ‖ · ‖H is nondegenerate [Ho90],[LM95] proceed by proving that, for A equal to a closed Darboux ball in M , there is a number cA > 0 such that any Hamiltonian diffeomorphism φ such that φ(A)∩ A= ∅ must have ‖φ‖H ≥ cA; if (M ,ω) is geometrically bounded [Che98] establishes a similar bound with A instead equal to a compact Lagrangian submanifold of M , generalizing an earlier result of [P93]. Along similar lines, if A is a compact Lagrangian submanifold of a tame symplectic manifold (M ,ω) and if U is either an open set intersecting A or a compact Lagrangian submanifold that intersects A transversely (and nontrivially) then there is a number cA,U > 0 such that one has the bound ‖φ‖H ≥ cA,U whenever φ(A)∩Ū = ∅. (This is [U14, Theorem 4.9]; see also [BC06, Corollary 3.7], [FOOO09, Theorem J], and [Cha12] for results which cover less general situations but have stronger bounds cA,U .) Recently, similar results to some of those above have appeared in [BM13, Theorem 1.5(ii)] and in [HLS13, Lemma 9, citing [LR]], but with a surprising twist. For certain rather specific classes of symplectic manifolds (M ,ω) and Lagrangian submanifolds A, for any open set U intersecting A these authors produce a positive constant cA,U which serves as a lower bound not We might also mention the results [MVZ12, Lemma 2.1, Theorem 2.17(vi)] about Lagrangian spectral invariants, which in retrospect could be seen as anticipating this phenomenon.