Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings

Abstract

The Katok examples on $S^2$ are induced by Randers metrics obtained by perturbing the round metric by the standard rotational Killing field: allowing a scaling factor of the round metric gives a two-parameter family of Randers metrics $F_{α,β}$, parametrised by positive real numbers $(α,β)$. We compute the set of all $(α,β)\in (0,2+\sqrt{3})^2$ for which $D^*(S^2,F^*_{α,β})$, the unit codisc bundle with respect to $F^*_{α,β}$, symplectically embeds into the round codisc bundle $D^*S^2$. This set has the structure of an infinite staircase. We also establish a dictionary between embeddings of these codisc bundles, singular $A_1$-ellipsoid embeddings, and $\mathbb{Z} _2$-equivariant ellipsoid embeddings, showing that these embedding problems are equivalent. Furthermore, we prove the analogous results for the $\mathbb{R} P^2$ case and discuss a broader family of examples for which this dictionary applies.