Extend existence of two n-link trajectories between Lagrangian subspaces to all submanifolds

Determine whether the existence result in Theorem 3.12 extends to every immersed closed submanifold M ⊂ R^{2d}: specifically, establish that for any pair of transverse affine Lagrangian subspaces L1 and L2 ⊂ R^{2d} and for every integer n ≥ 1, there exist at least two distinct non-degenerate n-link outer symplectic billiard trajectories with reflection points on M connecting L1 to L2, without assuming condition (LL).

Background

The paper defines the outer symplectic billiard correspondence for an immersed closed submanifold M in the standard symplectic space R^{2d} and studies both periodic orbits and trajectories connecting two affine Lagrangian subspaces in general position. Using a variational setup with generating functions, the authors prove several existence results.

For trajectories connecting two Lagrangian subspaces, Theorem 3.12 shows that if M satisfies condition (LL)—meaning that for every point P ∈ R^{2d} there exists x ∈ M such that T_xM is not contained in the symplectic complement (x − P)^ω—then for every n ≥ 1 there are at least two distinct non-degenerate n-link outer symplectic billiard trajectories from L1 to L2. This condition is automatically satisfied, for instance, when dim M ≥ d. The introduction explicitly states that it is unknown whether this existence theorem holds for all submanifolds M without imposing (LL).