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PreprintCC BY 4.0arXiv

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

  • Nikolas Adaloglou
  • Johannes Hauber
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Abstract

The Katok examples on S2 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 Fα,β, parametrised by positive real numbers (α,β). We compute the set of all (α,β)∈ (0,2+√3)2 for which D*(S2,F*α,β), the unit codisc bundle with respect to F*α,β, symplectically embeds into the round codisc bundle D*S2. This set has the structure of an infinite staircase. We also establish a dictionary between embeddings of these codisc bundles, singular A1-ellipsoid embeddings, and ℤ 2-equivariant ellipsoid embeddings, showing that these embedding problems are equivalent. Furthermore, we prove the analogous results for the ℝ P2 case and discuss a broader family of examples for which this dictionary applies.

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Cite

BibTeXRIS
BibTeX
@misc{adaloglou2026quantitative,
  title = {{Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings}},
  author = {Nikolas Adaloglou and Johannes Hauber},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.35209},
  eprint = {2609.35209},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.35209},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Quantitative symplectic topology of Katok's examples and equivariant symplectic embeddings
AU  - Nikolas Adaloglou
AU  - Johannes Hauber
PY  - 2026
DA  - 2026/09/28
PB  - arXiv
DO  - 10.48550/arxiv.2609.35209
UR  - https://arxiv.org/abs/2609.35209
AB  - 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)∈ (0,2+√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 ℤ _2-equivariant ellipsoid embeddings, showing that these embedding problems are equivalent. Furthermore, we prove the analogous results for the ℝ P^2 case and discuss a broader family of examples for which this dictionary applies.
ER  -