Markov staircases
Abstract
Rational homology ellipsoids are certain Liouville domains diffeomorphic to rational homology balls and having Lagrangian pin-wheels as their skeleta. From the point of view of almost toric fibrations, they are a natural generalisation of usual symplectic ellipsoids. We study symplectic embeddings of rational homology ellipsoids into the complex projective plane and we show that for each Markov triple, this problem gives rise to an infinite staircase. A key ingredient in the proof is the result that any two such embeddings are Hamiltonian isotopic. We also prove constraints on sizes for pairs of disjoint embeddings.
Cite
@article{arxiv.2509.03224,
title = {Markov staircases},
author = {Nikolas Adaloglou and Joé Brendel and Jonny Evans and Johannes Hauber and Felix Schlenk},
journal= {arXiv preprint arXiv:2509.03224},
year = {2025}
}
Comments
61 pages, 23 figures; v2: Isotopy theorem extended to embeddings beyond the visible range, and its proof corrected and simplified. Also some references added