Hamiltonian knottedness and lifting paths from the shape invariant

2021/05/10 by Hind, Richard, Zhang, Jun
#53D12 #53D35 #FOS: Mathematics #Symplectic Geometry (math.SG)

paper · doi:10.48550/arxiv.2105.04526

Abstract

The Hamiltonian shape invariant of a domain X ⊂ \mathbb R⁴, as a subset of \mathbb R², describes the product Lagrangian tori which may be embedded in X. We provide necessary and sufficient conditions to determine whether or not a path in the shape invariant can lift, that is, be realized as a smooth family of embedded Lagrangian tori, when X is a basic 4-dimensional toric domain such as a ball B⁴(R), an ellipsoid E(a,b) with (b)/(a) ∈ \mathbb N_{≥ 2}, or a polydisk P(c,d). As applications, via the path lifting, we can detect knotted embeddings of product Lagrangian tori in many toric X. We also obtain novel obstructions to symplectic embeddings between domains that are more general than toric concave or toric convex.

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