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We consider the periodic defocusing cubic nonlinear Klein-Gordon equation inthree dimensions in the symplectic phase space $H^{\frac{1}{2}}(\mathbb{T}^3)\times H^{-\frac{1}{2}}(\mathbb{T}^3)$. This space is at the criticalregularity for this equation, and in this setting there is no globalwell-posedness nor any uniform control on the local time of existence forarbitrary initial data. We prove a local-in-time non-squeezing result and aconditional global-in-time result which states that uniform bounds on theStrichartz norms of solutions imply global-in-time non-squeezing. As aconsequence of the conditional result, we conclude non-squeezing for certainsubsets of the phase space, and in particular, we prove small datanon-squeezing for long times. The proofs rely on several approximation resultsfor the flow, which we obtain using a combination of probabilistic anddeterministic techniques.

Symplectic non-squeezing for the cubic nonlinear Klein-Gordon equation on $\mathbb{T}^3$