Markov staircases

TL;DR

<3-5 sentence high-level summary>The paper investigates symplectic embeddings of pin-ellipsoids E_{p,q}(\alpha,\beta) into CP^2, showing that for each Markov triple (p, p_2, p_3) with a companion q the embedding problem exhibits an infinite staircase structure governed by Markov mutations. The authors develop a robust framework using almost toric geometry, Vianna triangles, and pavilion blow-ups to translate embedding questions into combinatorial and holomorphic constraints, proving an isotopy theorem, constructing visible staircases, and establishing sharp two-ball packing inequalities. Central to the approach is a Hamiltonian-isotopy–based isotopy result and a non-orbifold proof of Evans–Smith type obstructions, together with a detailed analysis of regulation and broken rulings on pavilion blow-ups. These results connect Markov-number theory with symplectic embedding problems, providing explicit, geometry-driven obstructions and constructions and advancing understanding of how complex-analytic data control four-dimensional symplectic capacities and packings.

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.

Figures (23)

• Figure 1: The almost toric base diagram $\mathfrak{A}_{p,q}(\alpha,\beta)$ for $E_{p,q}(\alpha,\beta)$. The branch cut is parallel to the $(p,q)$-direction. The toric boundary comprises the two solid edges (not the top side), whose affine lengths are $\alpha$ and $\beta$; these are two segments of an unbroken straight line with respect to the integral affine structure on the base diagram.
• Figure 2: The beginning of the Markov graph, where we have omitted the repetitions at the first two Markov triples.
• Figure 3: The first three Vianna triangles (a) $\mathfrak{D}(1,1,1)$, (b) $\mathfrak{D}(2,1,1)$ and (c) $\mathfrak{D}(2,5,1)$ together with the first two visible pin-ellipsoids (shaded in (b) and (c)) for the $p=2,q=1$ staircase. These visible embeddings can be made to fill an arbitrarily large portion of the triangle; in case (c) you need to use a nodal slide to shrink the other branch cut.
• Figure 4: The staircase for $p=2$, $q=1$.
• Figure 5: The staircase for $p=5$, $q=1$.

Theorems & Definitions (132)

• Definition 1.2.1
• Remark 1.2.2
• Definition 1.2.3
• Remark 1.2.4
• Definition 1.3.1
• Definition 1.3.2: Companion numbers
• Theorem 1.4.1: Isotopy Theorem
• Corollary 1.4.3
• proof
• Remark 1.4.4