Discontinuous symplectic capacities

Symplectic Topology and Capacities

I am going to talk about symplectic topology, a field that has seen a remarkable development in the past 15 years. Let’s begin with what was classically known – which was rather little. We start with a symplectic form ω, that is, a closed 2-form which is nondegenerate. This last condition means that ω is defined on an even dimensional manifold M and that its wedge with itself n times is a top d...

Symplectic Capacities of Domains in C

On Symplectic Capacities and Volume Radius

In this work we present an improvement to a theorem by C. Viterbo, relating the symplectic capacity of a convex body and its volume. This provides one more step towards the proof of the following conjecture: among all convex bodies in R 2n with a given volume, the Euclidean ball has maximal symplectic capacity. More precisely, the conjecture states that the best possible constant γn such that f...

Hofer’s question on intermediate symplectic capacities

Roughly twenty five years ago Hofer asked: can the cylinder B2(1)×R2(n−1) be symplectically embedded into B2(n−1)(R)×R2 for some R > 0? We show that this is the case if R > √ 2n−1 + 2n−2 − 2. We deduce that there are no intermediate capacities, between 1-capacities, first constructed by Gromov in 1985, and n-capacities, answering another question of Hofer. In 2008, Guth reached the same conclus...

An Example for Nonequivalence of Symplectic Capacities

We construct an open bounded star-shaped set Ω ⊂ R 4 whose cylindrical capacity is strictly bigger than its proper displacement energy.