On the Existence of Symplectic Barriers

Results concerning rigidity of Lagrangian submanifolds lie at the heart of symplectic topology, and have been intensively studied since the 1990s. An example for this phenomenon is the concept of Lagrangian Barriers, a form of symplectic rigidity introduced by Biran in 2001, which involves obligatory intersections of symplectic embeddings with Lagrangian submanifolds not derived from mere topology. In this joint work with Richard Hind and Yaron Ostrover, we present what appears to be the first illustration of Symplectic Barriers (and in particular not Lagrangian). The key point being that Lagrangian submanifolds are not the sole barriers, and there exist situations where a symplectic submanifold does not exhibit flexibility. In our work, we also tackle a question by Sackel–Song–Varolgunes–Zhu and provide bounds on the capacity of the ball after removing a codimension 2 hyperplane with a prescribed Kähler angle.