FOUR-PERIODIC INFINITE STAIRCASES FOR FOUR-DIMENSIONAL POLYDISKS CADEN FARLEY TARA S. HOLM NICKI MAGILL JEMMA SCHRODER ZICHEN WANG

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FOUR-PERIODIC INFINITE STAIRCASES FOR

FOUR-DIMENSIONAL POLYDISKS

CADEN FARLEY, TARA S. HOLM, NICKI MAGILL, JEMMA SCHRODER, ZICHEN WANG,

MORGAN WEILER, AND ELIZAVETA ZABELINA

Abstract.

The ellipsoid embedding function of a symplectic four-manifold measures

the amount by which its symplectic form must be scaled in order for it to admit an

embedding of an ellipsoid of varying eccentricity. This function generalizes the Gromov

width and ball packing numbers. In the one continuous family of symplectic four-

manifolds that has been analyzed, one-point blowups of the complex projective plane,

there is an open dense set of symplectic forms whose ellipsoid embedding functions are

completely described by ﬁnitely many obstructions, while there is simultaneously a

Cantor set of symplectic forms for which an inﬁnite number of obstructions are needed.

In the latter case, we say that the embedding function has an inﬁnite staircase. In

this paper we identify a new inﬁnite staircase when the target is a four-dimensional

polydisk, extending a countable family identiﬁed by Usher in 2019. Our work computes

the function on inﬁnitely many intervals and thereby indicates a method of proof for a

conjecture of Usher.

Contents

1. Introduction 2

1.1. Summary of results 4

1.2. Connections to other targets 8

1.3. Outline of the paper 9

Acknowledgements 9

2. Tools for obstructing and constructing embeddings 9

2.1. Embedding functions of toric domains 9

2.2. Quasi-perfect Diophantine classes 12

2.3. ECH capacities 14

2.4. Almost toric ﬁbrations 17

2.5. Combining obstructions and embeddings 20

3. Proof of the main theorem 21

3.1. Outer corners 24

3.2. Inner corners 28

4. Other properties of the embedding function 46

4.1. Towards Conjecture 1.1.2 46

4.2. Usher’s Conjecture 47

4.3. Descending staircases and fractal structure 48

4.4. Brahmagupta moves 49

5. Code for exploring ATFs 50

Date: August 30, 2023.

arXiv:2210.15069v3 [math.SG] 28 Aug 2023

2 FARLEY, HOLM, MAGILL, SCHRODER, WANG, WEILER, ZABELINA

References 55

1. Introduction

Asymplectic form on a 2

-dimensional smooth manifold

is a diﬀerential 2-form

satisfying:

•dω = 0, i.e., ωis closed, and

•ωn̸= 0, i.e., ωis nondegenerate.

A symplectic form can be thought of as a skew-symmetric version of a Riemannian

metric, providing area rather than length measurement. Symplectic geometry forms the

mathematical framework for classical mechanics and is a go-between from Riemannian

to complex geometry.

The volume

vol

(

)of a symplectic manifold is the quantity

RXωn

. We say a smooth

embedding

: (

X, ω

)

→

(

X′, ω′

)is symplectic if

ϕ∗

(

ω′

) =

, and we denote symplectic

embedding by

ϕ: (X, ω)s

→(X′, ω′),

→X′

when the symplectic form is clear from context and we are not emphasizing

the speciﬁc embedding ϕ.

Let (

X, ω

)be a four-dimensional symplectic manifold. Its ellipsoid embedding

function1is

cX(z) := inf nλ(E(1, z), ω0)s

→(X, λω)o,(1.0.1)

where

z∈R>0

λX

:= (

X, λω

)is

with the symplectic form scaled, the ellipsoid

E(c, d)⊂C2is the set

E(c, d) = (ζ1, ζ2)∈C2π|ζ1|2

c+|ζ2|2

d<1,

and

ω0

is the standard symplectic form

dx1∧dy1

dx2∧dy2

. Note that the

associated volume form is twice the standard volume form on

, thus

vol

(

c, d

)) =

There is a symmetry that allows us to reduce to

z≥

1. Namely, for 0

< z <

1we have

(

) =

zcX

), because

ω0

restricted to

, z

)equals

zω0

restricted to

/z,

under the diﬀeomorphism (

ζ1, ζ2

)

7→

(

ζ1/√z, ζ2/√z

). Therefore, from now on we restrict

the domain of cX(z)to R≥1.

The ellipsoid embedding function generalizes the Gromov width2via

cGr(X, ω) = 1

cX(1)

and the fraction of the volume of

that can be ﬁlled by

n∈Z≥1

equal balls can, by

[Mc1, Thm. 1.1], be computed from cXvia

cX(n)2vol(X).

1It is sometimes also called the embedding capacity function or capacity function.

The Gromov width of a symplectic manifold is

sup {r|E

(

r, r

)

→

(

X, ω

)

}

, or the largest ball that

embeds into (X, ω).

FOUR-PERIODIC INFINITE STAIRCASES FOR FOUR-DIMENSIONAL POLYDISKS 3

For a class of targets (

X, ω

)called “ﬁnite type convex toric domains” (see §2.1) which

includes the polydisks that we study, the ellipsoid embedding function satisﬁes several

key properties.

Proposition 1.0.1 ([

CGHMP

, p. 4, Prop. 2.1]).Let (

X, ω

)be a ﬁnite type convex toric

domain. The ellipsoid embedding function cX(z)satisﬁes the following properties.

(i) cX(z)≥qz

vol(X);

(ii) cXis nondecreasing;

(iii) cXis sublinear: for all t≥1, we have cX(tz)≤tcX(z);

(iv) cX(z)is continuous (in z);

(v) cX(z)is equal to the volume curve for suﬃciently large values of z; and

(vi) cX

(

)is piecewise linear, when not equal to the volume curve and not at the limit

of singular points.

We say

has an inﬁnite staircase if it is nonsmooth at inﬁnitely many points.

An outer corner is a nonsmooth point near which the function is concave while an

inner corner is is one near which the function is convex. By Proposition 1.0.1 (v), the

set of nonsmooth points is bounded. By [

CGHMP

, Thm. 1.13] (see Theorem 2.1.2 for

a statement in our case), the nonsmooth points of

have a unique ﬁnite limit point

called the accumulation point, whose

-coordinate we denote by

acc

(

). (By abuse

of notation, we also refer to this

-coordinate as the “accumulation point.”) We say an

inﬁnite staircase is ascending if the nonsmooth points accumulate from the left and

descending if the nonsmooth points accumulate from the right. These concepts are

illustrated in Figure 1.0.2. In this paper, we will establish the existence of an ascending

staircase.

1234567

0.5

1.5

2.5

. . .

Figure 1.0.2. In blue, the graph of the embedding capacity function

for a ball

(1) is shown on the domain indicated. The graph in red

is the volume lower bound established in Proposition 1.0.1(i). The point

marked O is an outer corner and the point marked I is an inner corner.

This target has an ascending inﬁnite staircase, ﬁrst identiﬁed by McDuﬀ

and Schlenk [

McSc

] and called the Fibonacci staircase in the literature.

The green point is the accumulation point.

4 FARLEY, HOLM, MAGILL, SCHRODER, WANG, WEILER, ZABELINA

1.1. Summary of results. Our target of choice will be the polydisk, deﬁned for

β∈R≥1by

P(1, β) := (ζ1, ζ2)∈C2π|ζ1|2≤1, π|ζ2|2≤β.

We denote by

cβ

its ellipsoid embedding function

cP(1,β)

. The polydisk is a ﬁnite type

convex toric domain, so

cβ

satisﬁes Proposition 1.0.1. In this case there are two functions

acc(β) := acc(P(1, β)) : [1,∞)→h3+2√2,∞

vol(β) := sacc(β)

vol(P(1, β)) : [1,∞)→"1 + √2

2,1!

where if

cβ

has an inﬁnite staircase, its accumulation point has coordinates (

acc

(

)

,vol

(

))

by [CGHMP, Thm. 1.13]; see Lemma 2.1.3.

The ﬁrst ellipsoid embedding function was computed for

1) by McDuﬀ

and Schlenk in [

McSc

]. They found that its graph contained an inﬁnite staircase whose

inner and outer corners were derived from the Fibonacci numbers. Further work by

Frenkel and Müller in [

] exhibited a similar inﬁnite staircase in

governed by the

Pell numbers, while on the other hand work of Cristofaro-Gardiner, Frenkel, and Schlenk

showed that the property of having an inﬁnite staircase is not universal: the functions

for

n∈Z>1

do not contain inﬁnite staircases [

CGFS

]. More generally, a conjecture of

Cristofaro-Gardiner, Holm, Mandini, and Pires in [

CGHMP

] suggests that

cβ

should not

contain an inﬁnite staircase for any rational β.

However, work by Usher [

] suggested that the set of irrational

for which

cβ

has

an inﬁnite staircase might be quite rich: he identiﬁed a bi-inﬁnite family

Ln,k ∈R≥1

for

which cLn,k have inﬁnite staircases.3Of particular interest to us are his

Ln,0:= pn2−1, n ≥2

which generate the

k >

0values of

with inﬁnite staircases (see §4.4). See Figure 1.1.1

for a visualization of these results via a plot of the relevant accumulation points.

In this paper as well as in the closely related papers [

BHM

], [

], and [

MMW

] we use

to denote

the staircase step and

to denote the image of a step, staircase, or

value under a symmetry analogous

to Usher’s Brahmagupta moves ([

, Def. 2.10]). Our notation diﬀers from Usher’s in that what the

and

indices denote are switched. We generally stick to our convention throughout but use Usher’s

convention here.

FOUR-PERIODIC INFINITE STAIRCASES FOR FOUR-DIMENSIONAL POLYDISKS 5

5 6 7 8 9 10 11 12 13

1.25

1.5

1.75

Figure 1.1.1. This ﬁgure shows the parameterized curve (

acc

(

)

,vol

(

))

in red. The point on the curve at

represents a point at which an

inﬁnite staircase for

cβ

must accumulate, if it exists. The red dot is the

accumulation point of the Pell stairs of Frenkel-Müller; the blue dots are

the

Ln,0

staircases of Usher; and the black

s indicate values of

without

inﬁnite staircases, proved by Cristofaro-Gardiner–Frenkel–Schlenk. The

accumulation points of the new inﬁnite staircases of Theorem 1.1.1 and

Conjecture 1.1.2 are indicated by green dots.

Work by Bertozzi, Holm, Maw, McDuﬀ, Mwakyoma, Pires, and Weiler [

BHM

] and by

Magill and McDuﬀ [MM] proved an analogous result for the target

Hb:= (ζ1, ζ2)∈C2π|ζ1|2+π|ζ2|2≤1, π|ζ2|2≤1−b.

(The region

is equivalent in terms of ellipsoid embeddings, see §2.1.1, to

CP2

thus in the literature on inﬁnite staircases it is also called the Hirzebruch surface.)

They showed that there are two bi-inﬁnite families

bn,i,δ

, with

n, i ∈Z≥0

and

δ∈ {

}

for which

cHb

has an ascending inﬁnite staircase. Moreover, each ascending inﬁnite

staircase comes paired with a descending inﬁnite staircase.

One feature that all inﬁnite staircases described so far appear to have in common is

that their outer corners are at

-values whose continued fractions grow by a predictable

pattern of adding pairs of integers. Recall that real numbers can be described by their

continued fractions, e.g.

[m, n, ] = m+1

n+1

ℓ

with repeated parts denoted by

hm, {n, }ki= [m, n, 

|{z}

ktimes

],[m, {n, }∞] = [m, n, , n, , n, , . . . ].

Every positive real number has a continued fraction with all entries positive integers;

rational numbers have ﬁnite continued fractions, quadratic irrational numbers (irrational

roots of quadratic equations with rational coeﬃcients) have inﬁnite periodic continued

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FOUR-PERIODICINFINITESTAIRCASESFORFOUR-DIMENSIONALPOLYDISKSCADENFARLEY,TARAS.HOLM,NICKIMAGILL,JEMMASCHRODER,ZICHENWANG,MORGANWEILER,ANDELIZAVETAZABELINAAbstract.Theellipsoidembeddingfunctionofasymplecticfour-manifoldmeasurestheamountbywhichitssymplecticformmustbescaledinorderforittoadmitanembeddingo...

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