Journal of Symplectic Geometry

Volume 20 (2022)

Number 5

Epsilon-non-squeezing and $C^0$-rigidity of epsilon-symplectic embeddings

Pages: 1135 – 1158

DOI: https://dx.doi.org/10.4310/JSG.2022.v20.n5.a5

Author

Stefan Müller (Department of Mathematics, Stanford University, Stanford, California, U.S.A.; and Department of Mathematical Sciences, Georgia Southern University, Statesboro, Ga., U.S.A.)

Abstract

An embedding $\varphi : (M_1, \omega_1) \to (M_2, \omega_2)$ (of symplectic manifolds of the same dimension) is called $\epsilon$-symplectic if the difference $\varphi^\ast \omega_2 - \omega_1$ is $\epsilon$-small with respect to a fixed Riemannian metric on $M_1$. We prove that if a sequence of $\epsilon$-symplectic embeddings converges uniformly (on compact subsets) to another embedding, then the limit is $E$-symplectic, where the number $E$ depends only on $\epsilon$, and $E(\epsilon) \to 0$ as $\epsilon \to 0$. This generalizes $C^0$-rigidity of symplectic embeddings, and answers a question in topological quantum computing by Michael Freedman.

As in the symplectic case, this rigidity theorem can be deduced from the existence and properties of symplectic capacities. An $\epsilon$-symplectic embedding preserves capacity up to an $\epsilon$-small error, and linear $\epsilon$-symplectic maps can be characterized by the property that they preserve the symplectic spectrum of ellipsoids (centered at the origin) up to an error that is $\epsilon$-small. We also sketch an alternative proof using the shape invariant, which gives rise to an analogous characterization and rigidity theorem for $\epsilon$-contact embeddings.

Received 22 June 2018

Accepted 16 April 2022

Published 24 April 2023